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Big Ideas Math Algebra 1 Answers Chapter 1 Solving Linear Equations

Hello Students!! Are you looking for Big Ideas Math Algebra 1 Answer Key ? If our answer is true then your search is over. Here you can find Big Ideas Math Book Algebra 1 Solution Key Chapter 1 Solving Linear Equations for free. So, Students who feel solving linear equations as a tough chapter can make use of our page and learn simple methods or tricks to solve the questions. Solve Linear Equations with the help of Big Ideas Math Algebra 1 Chapter 1 Solving Linear Equations Answer Key.

Download Big Ideas Math Book Algebra 1 Chapter 1 Solving Linear Equations Answer Key for free of cost. The solutions seen in Chapter 1 Solving Linear Equations Big Ideas Math Algebra 1 Answer Key are prepared by the subject experts. You can find brief explanations for all the questions in Big Ideas Math Book Algebra 1 Common Core Chapter 1 Solving Linear Equations Solution.

Big Ideas Math Book Algebra 1 Answer Key Chapter 1 Solving Linear Equations

The topics covered in Big Ideas Math Book Algebra 1 Answer Key Chapter 1 are Solving Simple Equations, Multi-step Equations, Solving Equations with Variables on Both Sides, Solving Absolute Value Equations, and Rewriting Equations and Formulas. After practicing the problems from lessons and exercises students can test their knowledge by solving the problems presented in Quizzes, Chapter Tests, reviews, and Cumulative Assessments. It is very simple to get the Big Ideas Math Algebra 1 Answers Chapter 1 Solving Linear Equations just tap the links provided below and download BIM Algebra 1 Solution Key.

Solving Linear Equations Maintaining Mathematical Proficiency

Solving linear equations monitoring progress, lesson 1.1 solving simple equations, solving simple equations 1.1 exercises, lesson 1.2 solving multi-step equations, solving multi-step equations 1.2 exercises, lesson 1.3 solving equations with variables on both sides, solving equations with variables on both sides 1.3 exercises, solving linear equations study skills: completing, solving linear equations 1.1-1.3 quiz, lesson 1.4 solving absolute value equations, solving absolute value equations 1.4 exercises, lesson 1.5 rewriting equations and formulas.

• Rewriting Equations and Formulas 1.5 Exercises

Solving Linear Equations Chapter Review

Solving linear equations chapter test, solving linear equations cumulative assessment.

Add or subtract.

Question 1. -5 + (-2) Answer: -5 + (-2 ) = -7

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -5 + ( -2 ) = -5 – 2 = – ( 5 + 2 ) = -7 Hence, from the above, We can conclude that, -5 + ( -2 ) = -7

Question 2. 0 + (-13) Answer: 0 + -13 = -13

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 0 + ( -13 ) = 0 – 13 = -13 Hence, from the above, We can conclude that, 0 + ( -13 ) = -13

Question 3. -6 + 14 Answer: -6 + 14 = 8

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -6 + 14 = +14 – 6 = +8 = 8 Hence, from the above, We can conclude that -6 +14 = 8

Question 4. 19 – (-13) Answer: 19 – ( -13 ) = 32

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 19 – ( -13 ) = 10 + 9 + 10 + 3 = 20 + 12 = 32 Hence, from the above, We can conclude that, 19 – (-13 ) = 32

Question 5. -1 – 6 Answer: -1 -6 = -7

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -1 – 6 = – ( 1 + 6 ) = -7 Hence, from the above, We can conclude that -1 -6 = -7

Question 6. – 5 – (-7) Answer: -5 – ( -7 ) = 2

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -5 – ( -7 ) = -5 + 7 = 7 – 5 = 2 Hence, from the above, We can conclude that -5 – ( -7 ) = 2

Question 7. 17 + 5 Answer: 17 + 5 = 22

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 17 + 5 = 15 + 2 + 5 = 22 Hence, from the above, We can conclude that 17 + 5 = 22

Question 8. 8 + (-3) Answer: 8 + ( -3 ) = 5

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 8 + ( -3 ) = 8 – 3 = 5 Hence, from the above, We can conclude that 8  + ( -3 ) = 5

Question 9. 11 – 15 Answer: 11 – 15 = -4

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 11 – 15 = -15 + 11 = – ( 15 – 11 ) = -4 Hence, from the above, We can conclude that, 11 – 15 = -4

Multiply or divide.

Question 10. -3(8) Answer: -3(8) = -3 × 8 = -24

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -3(8) = -3 × +8 = -24 Hence, from the above, We can conclude that -3 ( 8 ) = -24

Question 11. -7 • (-9) Answer: -7 . ( -9 ) = -7 × +9 = -63

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -7 . ( 9 ) = -7 × +9 = -63 Hence, from the above, We can conclude that -7 . ( 9 ) = -63

Question 12. 4 • (-7) Answer: 4 . ( -7 ) = 4 × ( -7 ) = -28

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 4. (-7 ) = +4 × -7 = -28 Hence, from the above, We can conclude that 4 . ( -7 ) = -28

Question 13. -24 ÷ (-6) Answer: -24 ÷ ( -6 ) = 4

Explanation: We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -24 ÷ ( -6 ) = + ( 24 ÷ 6 ) = 4 Hence, from the above, We can conclude that -24 ÷ ( -6 ) = 4

Question 14. -16 ÷ 2 Answer: -16 ÷ 2 = -8

Explanation: We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -16 ÷ 2 = -8 Hence, from the above, We can conclude that -16 ÷ 2 = -8

Question 15. 12 ÷ (-3) Answer: 12 ÷ ( -3 ) = -4

Explanation: We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 12 ÷ ( -3 ) = 12 ÷ -3 = -4 Hence, from the above, We can conclude that 12 ÷ ( -3 ) = -4

Question 16. 6 • 8 Answer: 6 . 8 = 6 × 8 = 48

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 6 . 8 = 6 × 8 = 48 Hence, from the above, We can conclude that 6 . 8 = 48

Question 17. 36 ÷ 6 Answer: 36 ÷ 6 = 6

Explanation: We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, 36 ÷ 6 = ( 30 + 6 ) ÷ 6 = ( 30 ÷ 6 ) + ( 6 ÷ 6 ) = 5 + 1 = 6 Hence, from the above, We can conclude that 36 ÷ 6 = 6

Question 18. -3(-4) Answer: -3 ( -4 ) = -3 × -4 = -12

Explanation: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – We know that, The result of any mathematical operation follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a So, -3 ( -4 ) = -3 × -4 = +12 Hence, from the above, We can conclude that -3 ( – 4 ) = 12

Question 19. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers, (c) multiplying integers, and (d) dividing integers. Give an example of each. Answer: a) Adding integers: We know that, The result of any mathematical operation i.e., Addition or subtraction follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a Example: Let take the two numbers -2 and 3 So, The addition of -2 and 3 is: -2 + 3 = +1 = 1 ( Since the big number has a positive sign )

b) Subtracting integers: We know that, The result of any mathematical operation i.e., Addition or subtraction follows the sign of the big number and if the two numbers have the same sign, then the result of that also operation also follows the same sign and if there is no sign before the number, then we can take that number as positive number i.e., +a Example: Let take the two numbers -3 and +8 So, The subtraction of -3 and +8 is: -3 – ( +8 ) = -3 – 8 = – ( 3 + 8 ) = -11 ( Since both the numbers have a negative sign )

c) Multiplying integers: We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – Example: Let take the two numbers +8 and -3 So, The multiplication of +8 and -3 is: +8 ( -3 ) = 8 × -3 = -24 ( Since + × – = – )

d) Dividing integers: We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – Example: Let take the two numbers -12 and -2 So, The division of -12 and -2 is: -12 ÷ ( -2 ) = -12 ÷ -2 = ( -10 + -2 ) ÷ -2 = ( -10 ÷ -2 ) + ( -2 ÷ -2 ) = 5 + 1 = 6 Hence, from the above, We can conclude all the rules for the four basic mathematical operations.

Solve the problem and specify the units of measure.

Question 1. The population of the United States was about 280 million in 2000 and about 310 million in 2010. What was the annual rate of change in population from 2000 to 2010? Answer: The annual rate of change in population from 2000 to 2010 is: 30 million

Explanation: It is given that the population of the United States was about 280 million in 2000 and about 310 million in 2010. So, The annual rate of change in population from 2000 to 2010 = ( The population of United States in 2010 ) – ( The population of United States in 2000 ) = 310 – 280 = 30 million Hence, from the above, We can conclude that the annual rate of change in population from 2000 to 2010 is: 30 million

Question 2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage (in miles per gallon)? Answer: Your car’s gas mileage ( in miles per gallon ) is: 30

Explanation: It is given that you drive 240 miles and use 8 gallons of gasoline. So, The mileage of your car = ( The total number of miles driven by your car ) ÷ ( The number of gallons of gasoline used by your car ) = 240 ÷ 8 = ( 160 + 80 ) ÷ 8 = ( 160 ÷ 8 ) + ( 80 ÷ 8 ) = 20 + 10 = 30 miles Hence, from the above, We can conclude that the mileage of your car is: 30 miles per gallon

Question 3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by 18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot per minute. About how long does it take for all the water to drain? Answer: The total time taken for the water to drain is: 2,430 minutes

Explanation: It is given that a bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by 18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic foot per minute. So, The volume of the rectangular prism = The dimensions of the rectangular prism = 5 × 3 × 18 × 12 = 3,240 cubic feet Now, The volume of the bathtub which is three-fourths full of water = \(\) {3}{4}[\latex] × 3240 = 2,430 cubic feet It is also given that the bathtub drains at a rate of 1 cubic foot per minute. So, The time taken to drain 2,430 cubic feet of water in minutes = 2,430 × 1 = 2,430 minutes Hence, from the above, We can conclude that the time taken for the water to drain from the bathtub at a rate of 1 cubic foot per minute is: 2,430 minutes

Essential Question How can you use simple equations to solve real-life problems?

Exploration 1 Measuring Angles

EXPLORATION 2 Making a Conjecture

EXPLORATION 3 Applying Your Conjecture

Communicate Your Answer

Question 4. How can you use simple equations to solve real-life problems? Answer: Any situation where there is an unknown quantity can be represented by a linear equation like calculating mileage rates and predicting profit. Example: Rahul is t years old and Ravi is 3 more than 8 times Rahul’s age. Their combined age is 39. How old is Rahul? t + (8t + 3) = 39 9t + 3 = 39 9t = 36 t = 36/9 t = 4. Therefore Rahul is old years old.

Monitoring Progress

Solve the equation. Justify each step. Check your solution.

Question 1. n + 3 = -7 Answer: The value of n is: -10

Explanation: The given equation is: n + 3 = -7 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, n + 3 = -7 n = -7 – (+3 ) n = -7 – 3 = -10 Hence from the above, We can conclude that the value of n is: -10

Question 2. g – \(\frac{1}{3}\) = –\(\frac{2}{3}\) Answer: The value of g is: –\(\frac{1}{3}\)

Explanation: The given equation is: g – \(\frac{1}{3}\) = –\(\frac{2}{3}\) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, g – \(\frac{1}{3}\) = –\(\frac{2}{3}\) g = –\(\frac{2}{3}\) + \(\frac{1}{3}\) g = \(\frac{-2 + 1}{3}\) g = \(\frac{-1}{3}\) g = –\(\frac{1}{3}\) Hence, fromthe above, We can conclude that the value of g is: –\(\frac{1}{3}\)

Question 3. -6.5 = p + 3.9 Answer: The value of p is: -10.4

Explanation: The given equation is: -6.5 = p + 3.9 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -6.5 = p + 3.9 p = -6.5 – 3.9 = – ( 6.5 + 3.9 ) = – 10.4 Hence, from the above, We can conclude that the value of p is: -10.4

Question 4. \(\frac{y}{3}\) = -6 Answer: The value of y is: -18

Explanation: The given equation is: \(\frac{y}{3}\) = -6 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, \(\frac{y}{3}\) = -6 \(\frac{y}{1}\) × \(\frac{1}{3}\) = -6 \(\frac{y}{1}\) = -6 ÷ \(\frac{1}{3}\) y = -6 × -3 y = -18 Hence, from the above, We can conclude that the value of y is: -18

Question 5. 9π = πx Answer: The value of x is: 9

Explanation: The given equation is: 9π = πx When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 9π = πx 9 × π = π × x x = ( 9 × π ) ÷ π x = 9 Hence, from the above, We can conclude that the value of x is: 9

Question 6. 0.05w = 1.4 Answer: The value of w is: 28

Explanation: The given equation is: 0.05w = 1.4 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 0.05w = 1.4 0.05 × w = 1.4 \(\frac{5}{100}\) × w = \(\frac{14}{10}\) w = \(\frac{14}{10}\) ÷ \(\frac{5}{100}\) w = \(\frac{14}{10}\) × \(\frac{100}{5}\) w = \(\frac{14 × 100}{10 × 5}\) w = \(\frac{28}{1}\) w = 28 Hence, from the above, We can conclude that the value of w is: 28

Question 7. Suppose Usain Bolt ran 400 meters at the same average speed that he ran the 200 meters. How long would it take him to run 400 meters? Round your answer to the nearest hundredth of a second. Answer: The time it took for him to run 400 meters is: 0.50 seconds

Explanation: It is given that Usain Bolt ran 400 meters at the same average speed that he ran the 200 meters. We know that, Speed = Distance ÷ Time But, it is given that the average speed is the same. Hence, Speed = Constant So, Since speed is constant, distance is directly proportional to time. So, The time taken by Usain Bolt to run 400 meters = 200 ÷ 400 = ( 2 × 100 ) ÷ ( 4 × 100 ) = 10 ÷ 20 = 0.50 seconds ( 0.5 and 0.50 are the same values Only for the representation purpose, we will add ‘0’ after 5 ) Hence from the above, We can conclude that the time is taken by Usain Bolt to run 400 meters when rounded-off to the nearest hundredth is: 0.50 seconds

Question 8. You thought the balance in your checking account was $68. When your bank statement arrives, you realize that you forgot to record a check. The bank statement lists your balance as $26. Write and solve an equation to find the amount of the check that you forgot to record. Answer: The amount of the check that you forgot to record is: $42

Explanation: It is given that you thought the balance in your checking account was $68 and when your bank statement arrives, you realize that you forgot to record a check and the bank statement lists your balance as $26. Now, Let the amount you forgot to record be: x So, The total balance in your checking account = ( The listed balance ) + ( The amount that you forgot to record a check ) 68 = 26 + x When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 68 = 26 + x x = 68 – 26 x = $42 Hence, from the above, We can conclude that the amount that forgot to record is: $42

Monitoring Progress and Modeling with Mathematics

In Exercises 5–14, solve the equation. Justify each step. Check your solution.

Question 1. VOCABULARY Which of the operations +, -, ×, and ÷ are inverses of each other? Answer: When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Hence, from the above, We can conclude that, + is inverse of –  and vice-versa × is inverse of ÷ and vice-versa

Question 2. VOCABULARY Are the equations -2x = 10 and -5x = 25 equivalent? Explain. Answer: The equations -2x = 10 and -5x = 25 are equivalent

Explanation: The given equations are: -2x = 10 and -5x = 25 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – So, From -2x = 10, x = 10 ÷ ( -2 ) x = -10 ÷ 2 x = -5 From -5x = 25, x = 25 ÷ ( -5 ) x = -25 ÷ 5 x = -5 Hence, from the above, We can conclude that the equations -2x = 10 and -5x = 25 are equivalent

Question 3. WRITING Which property of equality would you use to solve the equation 14x = 56? Explain. Answer: The given equation is: 14x = 56 So, It can be re-written as 14 × x = 56 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x = 56 ÷ 14 x = 4 Hence, from the above, We can conclude that the value of x is: 4

Explanation: Let the given equations be named as A), B), C), and D) So, The given equations are: A) 8 = x ÷ 2 B) 3 = x ÷ 4 C) x – 6 = 5 D) x ÷ 3 = 9 So, From the above equations, The equations A, B), and D) are dividing the numbers whereas equation C) subtracting the numbers Hence, from the above, We can conclude that, The equation C) does not belong to the other three.

Question 5. x + 5 = 8

Question 6. m + 9 = 2 Answer: The value of m is: -5

Question 7. y – 4 = 3

Question 8. s – 2 = 1 Answer: The value of s is: 3

Question 9. w + 3 = -4

Question 10. n – 6 = -7 Answer: The value of n is: -1

Question 11. -14 = p – 11

Question 12. 0 = 4 + q Answer: The value of q is: -4

Question 13. r + (-8) = 10

Question 14. t – (-5) = 9 Answer: The value of t is: 4

Answer: The equation for the original price is: p = x + $12.95

Explanation: It is given that a discounted amusement park ticket costs $12.95 less than the original price p. So, Let the discounted amusement park ticket be: x The given original price is: p So, The discounted amusement park ticket cost = p – $12.95 x = p – 12.95 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, p = x + $12.95 Hence, from the above, We can conclude that the equation for the original price is: p = x + $12.95

Answer: Your final score is: x = ( The score of your friend ) – 12

USING TOOLS The sum of the angle measures of a quadrilateral is 360°. In Exercises 17–20, write and solve an equation to find the value of x. Use a protractor to check the reasonableness of your answer.

Answer: The value of x is: 85 degrees

Explanation: We know that, The sum of angles in a quadrilateral is: 360 degrees So, 150 + 48 + 77 + x = 360 275 + x = 360 x = 360 – 275 x = 85 degrees Hence, from the above, We can conclude that the value of x is: 85 degrees

Answer: The value of x is: 100 degrees

Explanation: We know that, The sum of all angles in a quadrilateral is: 360 degrees So, 115 + 85 + 60 + x = 360 260 + x = 360 x = 360 – 260 x = 100 degrees Hence, from the above, We can conclude that the value of x is: 100 degrees

In Exercises 21–30, solve the equation. Justify each step. Check your solution.

Question 21. 5g = 20

Question 22. 4q = 52 Answer: The value of g is: 13

Explanation: The given equation is: 4g = 52 4 × g = 52 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, g = 52 ÷ 4 = ( 44 + 8 ) ÷ 4 = ( 44 ÷ 4 ) + ( 8 ÷ 4 ) = 11 + 2 = 13 Hence, from the above, We can conclude that the value of g is: 13

Question 23. p ÷ 5 = 3

Question 24. y ÷ 7 = 1 Answer: The value of y is: 7

Explanation: The given equation is: y ÷ 7 = 1 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, y = 1 × 7 y = 7 Hence, from the above, We can conclude that the value of y is: 7

Question 25. -8r = 64

Question 26. x ÷(-2) = 8 Answer: The value of x is: -16

Explanation: The given equation is: x ÷ ( -2 ) = 8 We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x ÷ ( -2 ) = 8 x = 8 × ( -2 ) x = -16 Hence, from the above, We can conclude that the value of x is: -16

Question 27. \(\frac{x}{6}\) = 8

Question 28. \(\frac{w}{-3}\) = 6 Answer: The value of w is: -18

Explanation: The given equation is: \(\frac{w}{-3}\) = 6 We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, \(\frac{w}{-3}\) = 6 w = 6 × ( -3 ) w = -18 Hence, from the above, We can conclude that the value of w is: -18

Question 29. -54 = 9s

Question 30. -7 = \(\frac{t}{7}\) Answer: The value of t is: -49

Explanation: The given equation is: -7 = \(\frac{t}{7}\) We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -7 = \(\frac{t}{7}\) t = -7 × 7 t = -49 Hence, from the above We can conclude that the value of t is: -49

In Exercises 31– 38, solve the equation. Check your solution.

Question 31. \(\frac{3}{2}\) + t = \(\frac{1}{2}\)

Question 32. b – \(\frac{3}{16}\) = \(\frac{5}{16}\) Answer: The value of b is: \(\frac{1}{2}\)

Explanation: The given equation is: b – \(\frac{3}{16}\) = \(\frac{5}{16}\) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, b = \(\frac{5}{16}\) + \(\frac{3}{16}\) Since the denominators of both the numerators are equal, add the numerators making the denominator common So, b = \(\frac{5 + 3}{16}\) b = \(\frac{8}{16}\) b = \(\frac{1}{2}\) Hence, from the above, We can conclude that the value of b is: \(\frac{1}{2}\)

Question 33. \(\frac{3}{7}\)m = 6

Question 34. –\(\frac{2}{5}\)y = 4 Answer: The value of y is: 10

Explanation: The given equation is: –\(\frac{2}{5}\)y = 4 We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, –\(\frac{2}{5}\)y = 4 –\(\frac{2}{5}\) × y = 4 y = 4 ÷ –\(\frac{2}{5}\) y = 4 × –\(\frac{5}{2}\) y = -4 × –\(\frac{5}{2}\) y = –\(\frac{4}{1}\) × –\(\frac{5}{2}\) y = –\(\frac{4 × 5}{1 × 2}\) y = 10 Hence, from the above, We can conclude that the value of y is: 10

Question 35. 5.2 = a – 0.4

Question 36. f + 3π = 7π Answer: The value of f is: 4π

Explanation: The given equation is: f + 3π = 7π When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, f + 3π = 7π f = 7π – 3π f = π ( 7 – 3 ) f = π ( 4 ) f = 4π Hence, from the above, We can conclude that the value of f is: 4π

Question 37. – 108π = 6πj

Question 38. x ÷ (-2) = 1.4 Answer: The value of x is: –\(\frac{14}{5}\)

Explanation: The given equation is: x ÷ ( -2 ) = 1.4 We know that, A) – ÷ – = + B) + ÷ – = – C) + ÷ + = + D) – ÷ + = – When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x ÷ ( -2 ) = 1.4 x ÷ ( -2 ) = \(\frac{14}{10}\) x ÷ ( -2 ) = \(\frac{7}{5}\) x = \(\frac{7}{5}\) × ( -2 ) x = – \(\frac{7}{5}\) × \(\frac{2}{1}\) x = –\(\frac{14}{5}\) Hence, from the above, We can conclude that the value of x is: –\(\frac{14}{5}\)

ERROR ANALYSIS In Exercises 39 and 40, describe and correct the error in solving the equation.

Answer: A negative 3 should have been multiplied to each side. We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – So, -(\(\frac{m}{3}\) ) =-4 -3  ( \(\frac{m}{3}\) ) = -4 ( -3 ) 3 ( \(\frac{m}{3}\) ) = -4 ( -3 ) 3 ( \(\frac{m}{3}\) ) = 12 \(\frac{m}{3}\)  × \(\frac{3}{1}\) = 12 m = 12 Hence, from the above, We can conclude that the value of m is: 12

MODELING WITH MATHEMATICS In Exercises 42– 44, write and solve an equation to answer the question.

Question 42. The temperature at 5 P.M. is 20°F. The temperature at 10 P.M. is -5°F. How many degrees did the temperature fall? Answer: The fall in temperature is: 25 degrees Fahrenheit

Explanation: It is given that the temperature at 5 P.M. is 20°F and the temperature at 10 P.M. is -5°F. So, The fall in temperature = ( The temperature at 5 P.M ) – ( The temperature at 10 P.M ) We know that, A) – × – = + B) + × – = – C) + × + = + D) – × + = – So, The fall in temperature = 20 – ( -5 ) = 20 + 5 = 25 degrees Fahrenheit Hence, from the above, We can conclude that the fall in temperature is: 25 degrees Fahrenheit

Question 44. The balance of an investment account is $308 more than the balance 4 years ago. The current balance of the account is $4708. What was the balance 4 years ago? Answer: The balance 4 years ago is: $4,400

Explanation: It is given that the balance of an investment account is $308 more than the balance 4 years ago. The current balance of the account is $4708. So, The current balance of the account = ( The balance of an investment account 4 years ago ) + $308 Let the balance of an investment account four years ago be x. When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, $4,708 = x + $308 x = 4,708 – 308 x = $4,400 Hence, from the above, We can conclude that the balance of an investment account four years ago is: $4,400

b. The length of a rectangular mat is twice the width. Use Guess, Check, and Revise to find the dimensions of one rectangular mat. Answer: The dimensions of the rectangular mat are: Length: 6 ft Width: 3 ft

Explanation: From the above problem, The area of the rectangular mat = 18 ft² It is given that the length of a rectangular mat is twice the width. We know that the area of the rectangle = ( Length ) × ( Width ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa It is also given that the length of a rectangular mat is twice the width So, Length of a rectangular mat = 2 × Width Now, The area of the rectangular mat = Length × Width 18 = 2 × Width × Width Width × Width = 18 ÷ 2 Width × Width = 9 From guessing, We can say that Width of the rectangular mat = 3 ft Now, The length of the rectangular mat = 2 × 3 = 6 ft Hence, from the above, We can conclude that the dimensions of the rectangular mat are: Length: 6 ft Width: 3 ft

Question 49. USING STRUCTURE Use the values -2, 5, 9, and 10 to complete each statement about the equation ax = b – 5. a. When a = ___ and b = ___, x is a positive integer. b. When a = ___ and b = ___, x is a negative integer.

b. How does the equation 7 + 9 + 5 + 48 + x = 100 relate to the circle graph? How can you use this equation to find the percent of cats sold? Answer: The percent of cats sold is: 31%

Explanation: We know that, In terms of percentages, any circle represents 100% So, The total percent of animals = The percent of animals that are represented by the circle 100% = 69% + x% When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x% = 100% – 69% x% = 31% Hence, from the above, We can conclude that the percent of cats is: 31%

Question 51. REASONING One-sixth of the girls and two-sevenths of the boys in a school marching band are in the percussion section. The percussion section has 6 girls and 10 boys. How many students are in the marching band? Explain.

Question 52. THOUGHT-PROVOKING Write a real-life problem that can be modeled by an equation equivalent to the equation 5x = 30. Then solve the equation and write the answer in the context of your real-life problem.

Answer: Let suppose there is some number of boys. The number of girls is five times of the boys and the total number of girls is 30. Find the number of boys? Ans: Let, The number of boys is x. It is given that the number of girls is five times of boys. So, The number of girls = 5x It is also given that The number of girls = 30 So, 5x = 30 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5 × x = 30 x = 30 ÷ 5 x = 6 Hence, from the above, We can conclude that the number of boys is: 6

MATHEMATICAL CONNECTIONS In Exercises 53–56, find the height h or the area of the base B of the solid.

Answer: The height of the cuboid is: 9 cm

Answer: The height of the prism is: \(\frac{5}{6}\) ft

Maintaining Mathematical Proficiency

Use the Distributive Property to simplify the expression.

Question 58. 8(y + 3) Answer: 8 ( y + 3 ) = 8y + 24

Explanation: The given expression is: 8 ( y + 3 ) We know that, By using the Distributive Property, a ( b + c ) = ( a × b ) + ( a × c ) So, By using the above Property, 8 ( y + 3 ) = ( 8 × y ) + ( 8 × 3 ) = 8y + 24 Hence, from the above, We can conclude that 8 ( y + 3 ) = 8y + 24

Question 60. 5(m + 3 + n) Answer: 5 ( m + 3 + n ) = 5m + 5n + 15

Explanation: The given expression is: 5 ( m + 3 + n ) By using the Distributive Property, a ( b + c ) = ( a × b ) + ( a × c ) So, By using the above Property, 5 ( m + 3 + n ) = ( 5 × m ) + ( 5 × 3 ) + ( 5 × n ) = 5m + 15 + 5n Hence, from the above, We can conclude that, 5 ( m + 3 + n ) = 5m + 15 + 5n

Question 61. 4(2p + 4q + 6)

Copy and complete the statement. Round to the nearest hundredth, if necessary.

Answer: The missing number is: \(\frac{1}{12}\)

Explanation: Let the missing number be: x So, The given equation is: \(\frac{5L}{min}\) = \(\frac{x L}{h}\) We know that, 1 hour = 60 minutes So, 1 min = \(\frac{1}{60}\) hour So, \(\frac{5 L}{min}\) = \(\frac{5 L × 1}{60h}\) \(\frac{5 L}{min}\) = \(\frac{1 L }{12h}\) So, x = \(\frac{1}{12}\) Hence, from the above, We can conclude that, The missing number is: \(\frac{1}{12}\)

Explanation: Let the missing number be: x So, The given equation is: \(\frac{7 gal}{min}\) = \(\frac{x qin}{sec}\) We know that, 1 min = 60 seconds 1 quintal = 100 kg 1 gallon = 3.78 kg = 4 kg So, 1 gallon = 0.04 quintal 1 sec = \(\frac{1}{60}\) min So, \(\frac{7 gal}{min}\) = \(\frac{x qin × 1}{60min}\) \(\frac{7 gal}{min}\) = \(\frac{1 L }{12h}\) So, x = \(\frac{1}{12}\) Hence, from the above, We can conclude that, The missing number is: \(\frac{1}{12}\)

Essential Question

How can you use multi-step equations to solve real-life problems?

EXPLORATION 1 Solving for the Angle Measures of a Polygon

Question 3. How can you use multi-step equations to solve real-life problems? Answer: A multi-step equation is an equation that takes two or more steps to solve. These problems can have a mix of addition, subtraction, multiplication, or division. We also might have to combine like terms or use the distributive property to properly solve our equations

Question 4. In Exploration 1, you were given the formula for the sum S of the angle measures of a polygon with n sides. Explain why this formula works. Answer: We know that, The sum of the angles in a triangle is: 180 degrees The triangle is also a quadrilateral So, A quadrilateral can be formed by the minimum of the three lines So, The minimum sum of all the angles in a quadrilateral is: 180 degrees Now, Let suppose we form a quadrilateral with 4 sides. So, The sum of all the angles in a quadrilateral = 360 degrees = 180 degrees × 2 = 180 degrees ( 4 sides -2 ) Let suppose we form a quadrilateral with 5 sides So, The sum of all the angles in a quadrilateral = 540 degrees = 180 degrees × 3 = 180 degrees ( 5 -2 ) Hence, in general, We can conclude that the sum of all the angles with n sides in a quadrilateral = 180 degrees ( n-2 )

Question 5. The sum of the angle measures of a polygon is 1080º. How many sides does the polygon have? Explain how you found your answer. Answer: The number of sides the polygon with 1080° have: 6

Explanation: It is given that the sum of all angle measures of a polygon is: 1080° We know that, The sum of angle measures with n sides in a polygon = 180° ( n – 2 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 1080° =180° ( n – 2 ) n – 2 = 1080 ÷ 180 n – 2 = 6 n = 6 + 2 n = 8 Hence, from the above, We can conclude that the number of sides of the polygon with sum of the angles 1080° is: 6

Solve the equation. Check your solution.

Question 1. -2n + 3 = 9 Answer: The value of n is: -3

Explanation: The given equation is: -2n + 3 = 9 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -2n + 3 = 9 -2n = 9 – (+3 ) n = 6 ÷ ( -2 ) = -3 Hence from the above, We can conclude that the value of n is: -3

Question 2. -21 = \(\frac{1}{2}\) – 11

Question 3. -2x – 10x + 12 = 18 Answer: The value of x is: –\(\frac{1}{2}\)

Explanation: The given equation is: -2x – 10x + 12 = 18 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -( 2x + 10x ) = 18 – 12 -12x = 6 x = 6 ÷ ( -12 ) x = –\(\frac{1}{2}\) Hence, from the above, We can conclude that the value of x is: –\(\frac{1}{2}\)

Question 4. 3(x + 1) + 6 = -9 Answer: The value of x is: -6

Explanation: The given equation is: 3 ( x + 1 ) + 6 = -9 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 3 ( x + 1 ) = -9 – (+6 ) By using the Distributive property, 3 ( x + 1 ) = 3x + 3 So, 3x + 3 = -15 3x = -15 – ( +3 ) 3x = -18 x = -18 ÷ 3 x = -6 Hence, from the above, We can conclude that the value of x is: -6

Question 5. 15 = 5 + 4(2d – 3) Answer: The value of d is:\(\frac{11}{4}\)

Explanation: The given equation is: 15 = 5 + 4 ( 2d – 3 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa 4 ( 2d – 3 ) = 15 -5 4 ( 2d – 3 ) = 10 By using the Distributive property, 4 ( 2d – 3 ) = 4 (2d ) -4 (3 ) = 8d – 12 So, 8d – 12 = 10 8d = 10 + 12 8d = 22 d = 22 ÷ 8 d = \(\frac{11}{4}\) Hence, from the above, We can conclude that the value of d is: \(\frac{11}{4}\)

Question 6. 13 = -2(y – 4) + 3y Answer: The value of y is: 5

Explanation: The given equation is: 13 = -2 ( y – 4 ) + 3y When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa By using the Distributive Property, -2 ( y – 4 ) = -2y + 8 So, 13 = -2y + 8 + 3y 13 = y + 8 y = 13 – 8 y = 5 Hence, from the above, We can conclude that the value of y is: 5

Question 7. 2x(5 – 3) – 3x = 5 Answer: The value of x is: 5

Explanation: The given equation is: 2x ( 5 – 3 ) – 3x = 5 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2x ( 2 ) – 3x = 5 4x – 3x = 5 x = 5 Hence, from the above, We can conclude that the value of y is: 5

Question 8. -4(2m + 5) – 3m = 35 Answer: The value of m is: -5

Explanation: The given equation is: -4 ( 2m + 5 ) – 3m = 35 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Now, By using the Distributive Property, -4 ( 2m + 5 ) = -4 (2m ) + 5 ( -4 ) = -8m -20 So, -8m -20 -3m = 35 -11m – 20 = 35 -11m = 35 + 20 -11m = 55 m = 55 ÷ ( -11 ) m = -5 Hence, from the above, We can conclude that the value of m is: -5

Question 9. 5(3 – x) + 2(3 – x) = 14 Answer: The value of x is: 1

Explanation: The given equation is: 5 ( 3 – x ) + 2 ( 3 – x ) = 14 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Now, By using the Distributive property, 5 ( 3 – x ) = 5 (3 ) -5 (x) = 15 – 5x 2 ( 3 – x ) = 2 (3) – 2 ( x) = 6 – 2x So, 15 – 5x + 6 – 2x = 14 21 – 7x = 14 7x = 21 – 14 7x = 7 x = 7 ÷ 7 x = 1 Hence, from the above, We can conclude that the value of x is: 1

Explanation: It is given that the formula d = \(\frac{1}{2}\)n + 26 relates the nozzle pressure n (in pounds per square inch) of a fire hose and the maximum horizontal distance the water reaches d (in feet). So, The given equation is: d = \(\frac{1}{2}\)n + 26 Where, d is the maximum horizontal distance n is the pressure It is also given that the maximum horizontal distance is: 50 feet So, 50 = \(\frac{1}{2}\)n + 26 \(\frac{1}{2}\)n = 50 – 26 \(\frac{1}{2}\)n = 24 \(\frac{1}{2}\) × n = 24 n = 24 × 2 n = 48 pounds per square inch Hence, from the above We can conclude that the pressure needed to reach 50 feet away is: 48 pounds per square inch

Explanation; It is given that you  have 96 feet of fencing to enclose a rectangular pen for your dog. To provide sufficient running space for your dog to exercise, the pen should be three times as long as it is wide. So, The perimeter of the rectangular pen is: 96 feet We know that, The perimeter of the rectangle = 2 (Length + Width ) It is also given that the pen is three times as long as it is wide So, Width = 3 × Length So, The perimeter of the rectangular pen =2 (  Length + ( 3 × Length ) ) 96 = 2 ( 4 × Length ) 4 × Length = 96 ÷ 2 4 × Length = 48 Length = 48 ÷ 4 Length = 12 feet So, Width = 3 × Length = 3 × 12 = 36 feet hence, from the above, We can conclude that The dimensions of the rectangular pen are: Length of the pen is: 12 feet Width of the pen is: 36 feet

In Exercises 3−14, solve the equation. Check your solution.

Vocabulary and Core ConceptCheck

Question 1. COMPLETE THE SENTENCE To solve the equation 2x + 3x = 20, first combine 2x and 3x because they are _________. Answer: The given equation is: 2x + 3x = 20 As 2x and 3x are combined by the symbol “+”, add 2x and 3x So, 2x + 3x = 5x So, 5x = 20 x = 20 ÷ 4 x = 5

Question 2. WRITING Describe two ways to solve the equation 2(4x – 11) = 10. Answer: The given equation is: 2 (4x – 11) = 10 Way-1: 2 × (4x – 11) = 10 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4x – 11 = 10 ÷ 2 4x – 11 = 5 4x = 5 + 11 4x = 16 x = 16 ÷ 4 x = 4 Hence, The value of x is: 4

Way-2: By using the Distributive Property, 2 (4x – 11) = 2 (4x) – 2 (11) = 8x – 22 So, 8x – 22 = 10 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 8x = 10 + 22 8x = 32 x = 32 ÷ 8 x = 4 Hence, The value of x is: 4

Question 3. 3w + 7 = 19

Question 4. 2g – 13 = 3 Answer: The value of g is: 8

Explanation: The given equation is: 2g – 13 = 3 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2g = 3 + 13 2g = 16 2 × g = 16 g = 16 ÷ 2 g = 8 Hence, from the above, We can conclude that the value of g is: 8

Question 5. 11 = 12 – q

Question 6. 10 = 7 – m Answer: The value of m is: -3

Explanation: The given equation is: 10 = 7 – m When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -m = 10 – 7 -m = 3 Multiply with “-” on both sides – (-m ) = -3 m = -3 Hence, from the above, We can conclude that the value of m is: -3

Question 7. 5 = \(\frac{z}{-4}\) – 3

Question 8. \(\frac{a}{3}\) + 4 = 6 Answer: The value of a is: 6

Explanation: The given equation is: \(\frac{a}{3}\) + 4 = 6 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, \(\frac{a}{3}\) = 6 – 4 \(\frac{a}{3}\) = 2 a = 2 × 3 a = 6 Hence, from the above, We can conclude that the value of a is: 6

Question 9. \(\frac{h + 6}{5}\) = 2

Question 10. \(\frac{d – 8}{-2}\) = 12 Answer: The value of d is: -16

Explanation: The given equation is: \(\frac{d – 8}{-2}\) = 12 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, d – 8 = 12 × (-2) d – 8 = -24 d = -24 + 8 d = -16 Hence, from the above, We can conclude that the value of d is: -16

Question 11. 8y + 3y = 44

Question 12. 36 = 13n – 4n Answer: The value of n is: 4

Explanation: The given equation is: 36 = 13n – 4n When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 36 = 9n 9n = 36 n = 36 ÷ 9 n = 4 Hence, from the above, We can conclude that the value of n is: 4

Question 13. 12v + 10v + 14 = 80

Question 14. 6c – 8 – 2c = -16 Answer: The value of c is: -2

Explanation: The given equation is: 6c – 8 – 2c = -16 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4c – 8 = -16 4c = -16 + 8 4 × c = -8 c = -8 ÷ 4 c = -2 Hence, from the above, We can conclude that the value of c is: -2

Question 16. MODELING WITH MATHEMATICS A repair bill for your car is $553. The parts cost $265. The labor cost is $48 per hour. Write and solve an equation to find the number of hours of labor spent repairing the car. Answer: The number of hours of labor spent repairing the car is: 6 hours

Explanation: It is given that a repair bill for your car is $553. The parts cost $265. The labor cost is $48 per hour. Let the number of hours of labor spent repairing the car be: x So, The total bill to repair your car = ( The labor cost per hour ) × ( The number of hours of labor spent repairing the car ) +  (The cost of the parts ) 553 = 48x + 265 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 48x = 553 – 265 48x = 288 48 × x = 288 x = 288 ÷ 48 x = 6 Hence, from the above, We can conclude that the number of hours of labor spent repairing the car is: 6 hours

In Exercises 17−24, solve the equation. Check your solution.

Question 17. 4(z + 5) = 32

Question 18. -2(4g – 3) = 3018. Answer: The value of g is: 378

Explanation: The given equation is: -2 (4g – 3) = 3018 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -2 × ( 4g – 3 ) = 3018 4g – 3 = 3018 ÷ 2 4g – 3 = 1,509 4g = 1,509 +3 4 × g = 1,512 g = 1,512 ÷ 4 g = 378 Hence, from the above, We can conclude that the value of g is: 378

Question 19. 6 + 5(m + 1) = 26

Question 20. 5h+ 2(11 – h) = -5 Answer: The value of h is: -9

Explanation: The given equation is: 5h + 2 ( 11-h ) = -5 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa By using the Distributive Property of Multiplication, 2 ( 11 – h ) = 2 (11 ) – 2 ( h ) = 22 – 2h So, 5h + 22 – 2h = -5 3h + 22 = -5 3h = -5 – (+22) 3h = -5 -22 3h = -27 h = -27 ÷ 3 h = -9 Hence, from the above, We can conclude that the value of h is: -9

Question 21. 27 = 3c – 3(6 – 2c)

Question 22. -3 = 12y – 5(2y – 7) Answer: The value of y is: -19

Explanation: The given equation is: -3 = 12y – 5 (2y – 7) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 12y – 5 (2y – 7) = -3 By using the Distributive Property of Multiplication, 5 ( 2y – 7 ) = 5 (2y ) – 5 (7 ) = 10y – 35 So, 12y – ( 10y – 35 ) = -3 12y – 10y + 35 = -3 2y + 35 = -3 2y = -3 – (+35 ) 2y = -3 – 35 2y = -38 y = -38 ÷ 2 y = -19 Hence, from the above, We can conclude that the value of y is: -19

Question 23. -3(3 + x) + 4(x – 6) = -4

Question 24. 5(r + 9) – 2(1 – r) = 1 Answer: The value of r is: -6

Explanation: The given equation is: 5 ( r + 9 ) – 2 ( 1 – r ) = 1 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Now, By using the Distributive Property of Multiplication, 5 ( r + 9 ) = 5 ( r ) + 5 ( 9 ) = 5r + 45 2 ( 1 – r ) = 2 ( 1 ) – 2 ( r ) = 2 – 2r So, 5r + 45 – ( 2 – 2r ) = 1 5r + 45 – 2 + 2r = 1 7r + 43 = 1 7r = 1 – 43 7r = -42 r = -42 ÷ 7 r = -6 Hence, from the above, We can conclude that the value of r is: -6

USING TOOLS In Exercises 25−28, find the value of the variable. Then find the angle measures of the polygon. Use a protractor to check the reasonableness of your answer.

In Exercises 29−34, write and solve an equation to find the number.

Question 29. The sum of twice a number and 13 is 75.

Question 30. The difference of three times a number and 4 is -19. Answer: The number is: -5

Explanation: It is given that the difference of three times of a number and 4 is -19 Now, Let the number be x So, The three times of a number = 3 (x) = 3x So, 3x – 4 = -19 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 3x = -19 + 4 3x = -15 x = -15 ÷ 3 x = -5 Hence, from the above, We can conclude that the number is: -5

Question 31. Eight plus the quotient of a number and 3 is -2.

Question 32. The sum of twice a number and half the number is 10. Answer: The number is: 4

Explanation: It is given that the sum of twice of a number and half the number is 10. Let the number be x. So, The twice of a number = 2 (x ) = 2x Half of the number = x ÷ 2 = \(\frac{x}{2}\) So, 2x + \(\frac{x}{2}\) = 10 2x can be rewritten as: \(\frac{4x}{2}\) So, \(\frac{4x}{2}\) + \(\frac{x}{2}\) = 10 \(\frac{4x + x}{2}\) = 10 \(\frac{5x}{2}\) = 10 5x = 10 × 2 5x = 20 x = 20 ÷ 5 x = 4 Hence, from the above, We can conclude that the numebr is: 4

Question 33. Six times the sum of a number and 15 is -42.

Question 34. Four times the difference of a number and 7 is 12. Answer: The number is: 4

Explanation: It is given that the four times the difference of a number and 7 is 12 Let the number be x So, Four times of the number = 4 ( x ) = 4x So, 4x – x = 12 3x = 12 x = 12 ÷ 3 x = 4 Hence, from the above, We can conclude that the number is: 4

USING EQUATIONS In Exercises 35−37, write and solve an equation to answer the question. Check that the units on each side of the equation balance.

Question 35. During the summer, you work 30 hours per week at a gas station and earn $8.75 per hour. You also work as a landscaper for $11 per hour and can work as many hours as you want. You want to earn a total of $400 per week. How many hours must you work as a landscaper?

Answer: The length d of the deep end is: 12 feet

Question 37. You order two tacos and a salad. The salad costs $2.50. You pay 8% sales tax and leave a $3 tip. You pay a total of $13.80. How much does one taco cost?

JUSTIFYING STEPS In Exercises 38 and 39, justify each step of the solution.

Answer: –\(\frac{1}{2}\) ( 5x – 8 ) – 1 = 6                          Write the equation –\(\frac{1}{2}\) ( 5x – 8 ) = 6 + 1                         Arrange the similar terms –\(\frac{1}{2}\) ( 5x – 8 ) = 7                                  Simplify – ( 5x – 8 ) = 7 × 2                                                              Divide by 2 on both sides – ( 5x – 8 ) = 14                                                                    Simplify 5x – 8 = -14                                                                       Multiply with “-” on both sides 5x = -14 + 8                                                                         Arrange the similar terms 5x = -6                                                                               Divide by 6 on both sides x = –\(\frac{6}{5}\)                                              The result Hence, The solution is: x = –\(\frac{6}{5}\)

ERROR ANALYSIS In Exercises 40 and 41, describe and correct the error in solving the equation.

Answer: The given equation is: -2 ( 7 – y ) + 4 = -4 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -2 ( 7 – y ) = -4 –  (+4 ) -2 ( 7 – y ) = -4 – 4 -2 ( 7 – y ) = -8 Now, By using the Distributive Property of Multiplication, 2 ( 7 – y ) = 2 ( 7 ) – 2 ( y ) = 14 – 2y So, – ( 14 – 2y ) = -8 2y – 14 = -8 2y = -8 + 14 2y = 6 y = 6 ÷ 2 y = 3 Hence, The value of y is: 3

MATHEMATICAL CONNECTIONS In Exercises 42−44, write and solve an equation to answer the question.

Answer: The dimensions of the court are: The Length of the court is: 36 feet The width of the court is: 78 feet

Answer: The length of each side is: 15 inches

Question 45. COMPARING METHODS Solve the equation 2(4 – 8x) + 6 = -1 using (a) Method 1 from Example 3 and (b) Method 2 from Example 3. Which method do you prefer? Explain.

Answer: Given that, The total cost for a football ticket order is $220.70 The cost of 1 ticket is $32.50. The convenience charge is $3.30 per ticket. The processing charge is 5.90 per ticket. The total price per 1 ticket = $32.50 + $3.30 + $5.90 = $41.7. For $220.70 number of tickets is $220.70/$41.7 = $5.2925. Therefore the number of tickets purchased is 5.

Question 47. MAKING AN ARGUMENT You have quarters and dimes that total $2.80. Your friend says it is possible that the number of quarters is 8 more than the number of dimes. Is your friend correct? Explain.

Question 49. REASONING An even integer can be represented by the expression 2n, where n is an integer. Find three consecutive even integers that have a sum of 54. Explain your reasoning.

b. Estimate the number of students who attended the fourth meeting. Answer: The number of students who attended the fourth meeting is: 24

Explanation: The mean attendance  of the first four meetings = ( The attendance of the 4 meetings ) ÷  ( The total number of meetings ) = ( 18 + 21 + 17 + x ) ÷ 4 It is given that the mean attendance of the first four meetings is: 20 So, 20 =  ( 18 + 21 + 17 + x ) ÷ 4 ( 56 + x ) ÷ 4 = 20 56 + x = 20 × 4 56 + x = 80 x = 80 – 56 x = 24 Hence, from the above, We can conclude that the number of students who attended the 4th meeting is: 24

c. Describe a way you can check your estimate in part (b). Answer: The estimate in part (b) can be checked by using the property of the mean So, The mean attendance of the four meetings = ( The attendance of the four meetings ) ÷ ( The total number of meetings ) = ( 18 + 21 + 17 + 24 ) ÷ 4 = 80 ÷ 4 = 20 Hence, from the above, We can conclude that the mean attendance of the four meetings is the same as given above.

REASONING In Exercises 51−56, the letters a, b, and c represent nonzero constants. Solve the equation for x.

Question 51. bx = -7

Question 52. x + a = \(\frac{3}{4}\) Answer: The value of x is: \(\frac{3}{4}\) – a

Explanation: The given equation is: x + a = \(\frac{3}{4}\) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x =  \(\frac{3}{4}\) – a Hence, from the above, We can conclude that the value of a is: \(\frac{3}{4}\) – a

Question 53. ax – b = 12.5

Question 54. ax + b = c Answer: The value of x is: \(\frac{c – b}{a}\)

Explanation: The given equation is: ax + b = c When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, ax = c – b x = \(\frac{c – b}{a}\) Hence, from the above, We can conclude that the value of x is: \(\frac{c – b}{a}\)

Question 55. 2bx – bx = -8

Question 56. cx – 4b = 5b Answer The value of x is: \(\frac{9b}{c}\)

Explanation: The given equation is: cx – 4b = 5b When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, cx = 5b + 4b cx = 9b x = \(\frac{9b}{c}\) Hence, from the above, We can conclude that the value of x is: \(\frac{9b}{c}\)

Simplify the expression.

Question 57. 4m + 5 – 3m

Question 58. 9 – 8b + 6b Answer: 9 – 8b + 6b = 9 – (8b – 6b ) = 9 – 2b

Question 59. 6t + 3(1 – 2t) – 5

Determine whether (a) x = −1 or (b) x = 2 is a solution of the equation.

Question 60. x – 8 = -9 Answer: x = -1 is a solution to the given equation

Explanation: The given equation is: x – 8 = -9 a) Let x = -1 So, -1 – 8 = -9 -9 = -9 As LHS is equal to RHS x = -1 is a solution of the given equation b) Let x = 2 So, 2 – 8 = -9 -6 = -9 As LHS is not equal to RHS, x = 2 is not a solution of the given equation

Question 61. x + 1.5 = 3.5

Question 62. 2x – 1 = 3 Answer: x = 2 is a solution to the given equation

Explanation: The given equation is: 2x – 1 = 3 a) Let x = -1 So, 2 ( -1 ) – 1 = 3 -2 – 1 = 3 -3 = 3 As LHS is not equal to RHS x = -1 is not a solution to the given equation b) Let x = 2 So, 2 ( 2 ) -1 = 3 4 – 1 = 3 3 = 3 As LHS is equal to RHS, x = 2  is a solution to the given equation

Question 63. 3x + 4 = 1

Question 64. x + 4 = 3x Answer: x = 2 is a solution to the given equation

Explanation: The given equation is: x + 4 = 3x a) Let x = -1 So, -1 + 4 = 3 ( -1 ) = 3 = -3 As LHS is not equal to RHS, x = -1 is not a solution to the given equation b) Let x = 2 So, 2 + 4 = 3 ( 2 ) 6 = 6 As LHS is equal to RHS, x = 2 is a solution to the given equation.

Question 65. -2(x – 1) = 1 – 3x

EXPLORATION 1 Perimeter

EXPLORATION 2 Perimeter and Area

Work with a partner.

• Each figure has the unusual property that the value of its perimeter (in feet) is equal to the value of its area (in square feet). Use this information to write an equation for each figure.
• Solve each equation for x. Explain the process you used to find the solution.
• Find the perimeter and area of each figure.

Question 3. How can you solve an equation that has variables on both sides? Answer: If the variable is the same on both sides in an equation, then rearrange the like terms So, Separate the variables and the numbers and simplify the variables and the numbers In this way, We can solve an equation with a single variable

Question 4. Write three equations that have the variable x on both sides. The equations should be different from those you wrote in Explorations 1 and 2. Have your partner solve the equations. Answer: Let the three equations that have variable x on both sides and different from Explorations 1 and 2 are: a) 6x + 2 = 5x-6 b) 16x = 18x – 2 c) 12x = 15x + 63 Now, a) The given equation is: 6x + 2 = 5x – 6 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 6x – 5x = -6 – 2 x = -8 Hence, The value of x is: -8 b) The given equation is: 9x = 18x – 2 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 18x – 16x = 2 2x = 2 x = 2 ÷ 2 x = 1 Hence, The value of x is: 1 c) The given equation is: 12x = 15x + 63 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 12x -15x = 63 -3x = 63 x = 63 ÷ ( -3 ) x = -63 ÷ 3 x = -21 Hence, The value of x is: -21

Question 1. -2x = 3x + 10 Answer: The value of x is: -2

Explanation: The given equation is: -2x = 3x + 10 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -2x – 3x = 10 -5x = 10 x = 10 ÷ (-5) x = -10 ÷ 5 x = -2 Hence, from the above, We can conclude that the value of x is: -2

Question 2. \(\frac{1}{2}\)(6h – 4) = -5h + 1 Answer: The value of h is: \(\frac{3}{8}\)

Explanation: The given equation is: \(\frac{1}{2}\) ( 6h – 4 ) = -5h + 1 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 6h – 4 = 2 ( -5h + 1 ) 6h – 4 = 2 ( -5h ) + 2 ( 1 ) [ By using the Distributive Property of Multiplication ) 6h – 4 = -10h + 2 6h + 10h = 2 + 4 16h = 6 h = \(\frac{6}{16}\) h= \(\frac{3}{8}\) Hence, from the above, We can conclude that the value of h is: \(\frac{3}{8}\)

Question 3. –\(\frac{3}{4}\)(8n + 12) = 3(n – 3) Answer: The value of n is: 0

Explanation: The given equation is: –\(\frac{3}{4}\) ( 8n + 12 ) = 3 ( n – 3 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 8n + 12 = –\(\frac{4}{3}\) × 3 ( n – 3 ) 8n + 12 = –\(\frac{4}{3}\) \(\frac{3}{1}\) ( n – 3 ) 8n + 12 = –\(\frac{3 × 4}{3 × 1}\) ( n – 3 ) 8n + 12 = -4 ( n – 3 ) 8n + 12 = -4n – 4 ( -3 ) 8n + 12 = -4n + 12 8n + 4n =12 – 12 12n = 0 n = 0 Hence, from the above, We can conclude that the value of n is: 0

Solve the equation.

Question 4. 4(1 – p) = 4p – 4 Answer: The value of p is: 1

Explanation: The given equation is: 4 ( 1 -p ) = 4p – 4 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4 ( 1 ) – 4 ( p ) = 4p – 4 4 – 4p = 4p – 4 4p + 4p = 4 + 4 8p = 8 p = 8 ÷ 8 p = 1 Hence, from the above, We can conclude that the value of p is: 1

Question 5. 6m – m = –\(\frac{5}{6}\)(6m – 10) Answer: The value of m is: \(\frac{5}{6}\)

Explanation: The given equation is: 6m – m = –\(\frac{5}{6}\) ( 6m – 10 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5m = –\(\frac{5}{6}\) ( 6m – 10 ) 5m = –\(\frac{5}{6}\) ( 6m ) – ( –\(\frac{5}{6}\) ( 10 ) ) 5m = –\(\frac{5}{6}\) × \(\frac{6m}{1}\) + \(\frac{5}{6}\) \(\frac{10}{1}\) 5m = –\(\frac{5 × 6m}{6 × 1}\) + \(\frac{5 × 10}{6 × 1}\) 5m = -5m + \(\frac{25}{3}\) 5m + 5m = \(\frac{25}{3}\) 10m = \(\frac{25}{3}\) m = \(\frac{25}{3}\) ÷ \(\frac{10}{1}\) m = \(\frac{25}{3}\) × \(\frac{1}{10}\) m = \(\frac{25}{30}\) m = \(\frac{5}{6}\) Hence, from the above, We can conclude that the value of m is: \(\frac{5}{6}\)

Question 6. 10k + 7 = -3 – 10k Answer: The value of k is: –\(\frac{1}{2}\)

Explanation: The given equation is: 10k + 7 = -3 – 10k When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 10k + 10k = -3 – ( +7 ) 20k = -3 – 20k = -10 k = -10 ÷ 20 k = –\(\frac{1}{2}\) Hence, from the above, We can conclude that the value of k is: –\(\frac{1}{2}\)

Question 7. 3(2a – 2) = -2(3a – 3) Answer: The value of a is: 1

Explanation: The given equation is: 3 ( 2a – 2 ) = -2 ( 3a – 3 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, By using the Distributive Property of Multiplication, 3 ( 2a ) – 3 ( 2 ) = -2 ( 3a ) + 2 ( 3 ) 6a – 6 = -6a + 6 6a + 6a = 6 + 6 12a = 12 a = 12 ÷ 12 a = 1 Hence, from the above, We can conclude that the value of a is: 1

Concept Summary

Question 8. A boat travels upstream on the Mississippi River for 3.5 hours. The return trip only takes 2.5 hours because the boat travels 2 miles per hour faster downstream due to the current. How far does the boat travel upstream? Answer: The distance the boat travel upstream is: 17.5 miles

Explanation: It is given that a boat travels upstream on the Mississippi River for 3.5 hours. The return trip only takes 2.5 hours because the boat travels 2 miles per hour faster downstream due to the current. Now, Let x be the speed of the boat traveled upstream We know that, Speed = Distance ÷ Time Distance = Speed × Time It is given that the time taken by the boat traveled upstream is: 3.5 hours So, Distance traveled upstream = 3.5 × x = 3.5x Now, It is also given that the speed of the boat is 2 miles per hour faster downstream So, Distance traveled downstream by boat = 2.5 ( x + 2 ) SO, As both the distances are the same, 3.5x = 2.5 ( x + 2 ) By using the Distributive Property of Multiplication, 3.5x = 2.5 ( x) + 2.5 ( 2 ) 3.5x = 2.5x + 5 3.5x – 2.5x = 5 x = 5 So, The distance traveled upstream by boat = 3.5x = 3.5 ( 5 ) = 17.5 miles per hour Hence, from the above, We can conclude that the distance traveled upstream by boat is: 17.5 miles per hour

In Exercises 3–16, solve the equation. Check your solution.

Question 1. VOCABULARY Is the equation -2(4 – x) = 2x + 8 an identity? Explain your reasoning. Answer: -2 ( 4 – x ) = 2x + 8 is not an identity

Explanation: The given equation is: -2 ( 4 – x ) = 2x + 8 By using the Distributive Property of Multiplication, -2 ( 4 ) + 2 ( x ) = 2x + 8 -8 + 2x = 2x + 8 2x – 8 = 2x + 8 As LHS ≠ RHS -2 ( 4 – x ) = 2x + 8  is not an identity

Question 2. WRITING Describe the steps in solving the linear equation 3(3x – 8) = 4x + 6 Answer: The value of x is: 6

Explanation: The given equation is: 3 ( 3x – 8 ) = 4x + 6 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, Now, By using the Distributive Property of Multiplication, 3 ( 3x ) – 3 ( 8 ) = 4x + 6 9x – 24 = 4x + 6 9x – 4x = 6 + 24 5x = 30 x = 30 ÷ 5 x = 6 Hence, from the above, We can conclude that the value of x is: 6

Question 3. 15 – 2x = 3x

Question 4. 26 – 4s = 9s Answer: The value of s is: 2

Explanation: The given equation is: 26 – 4s = 9s When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 26 = 9s + 4s 13s = 26 s = 26 ÷ 13 s = 2 Hence, from the above, We can conclude that the value of s is: 2

Question 5. 5p – 9 = 2p + 12

Question 6. 8g + 10 = 35 + 3g Answer: The value of g is: 5

Explanation: The given equation is: 8g + 10 = 35 + 3g When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 8g – 3g = 35 – 10 5g = 25 g = 25 ÷ 5 g = 5 Hence, from the above, We can conclude that the value of g is: 5

Question 7. 5t + 16 = 6 – 5t

Question 8. -3r + 10 = 15r – 8 Answer: The value of r is: 1

Explanation: The given equation is: -3r + 10 = 15r – 8 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -3r – 15r = -10 – 8 -18r = -18 r = -18 ÷ ( -18 ) r = 1 [ since – ÷ – = + ] Hence, from the above, We can conclude that the value of r is: 1

Question 9. 7 + 3x – 12x = 3x + 1

Question 10. w – 2 + 2w = 6 + 5w Answer: The value of w is: -4

Explanation: The given equation is: w – 2 + 2w = 6 + 5w When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, w + 2w -5w =6 + 2 -2w = 8 w = -8 ÷ 2 w = -4 Hence, from the above, We can conclude that the value of w is: -4

Question 11. 10(g + 5) = 2(g + 9)

Question 12. -9(t – 2) = 4(t – 15) Answer: The value of t is: 6

Explanation: The given equation is: -9 ( t – 2 ) = 4 ( t – 15 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, Now, By using the Distributive Property of Multiplication, -9 ( t ) +  9 ( 2 ) = 4 ( t ) – 4 ( 15 ) -9t + 18 = 4t – 60 -9t – 4t = -60 – 18 -13t = -78 t = -78 ÷ ( -13 ) t = 6 [ Since  -÷ – = + ] Hence, from the above, We can conclude that the value of t is: 6

Question 13. \(\frac{2}{3}\)(3x + 9) = -2(2x + 6)

Question 14. 2(2t + 4) = \(\frac{3}{4}\)(24 – 8t) Answer: The value of t is: 1

Explanation: The given equation is: 2 ( 2t + 4 ) = \(\frac{3}{4}\) ( 24 – 8t ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, By using the Distributive Property of Multiplication, 2 ( 2t ) + 2 ( 4 ) = \(\frac{3}{4}\) ( 24 ) – 8t  (\(\frac{3}{4}\) ) 4t + 8 = \(\frac{3}{4}\) × \(\frac{24}{1}\) – \(\frac{8t}{1}\) × \(\frac{3}{4}\) 4t + 8 = \(\frac{3 × 24}{4 × 1}\) – \(\frac{3 × 8t}{4 × 1}\) 4t + 8 = \(\frac{18}{1}\) – \(\frac{6t}{1}\) 4t + 8 = 18 – 6t 4t + 6t = 18 – 8 10t = 10 t = 10 ÷ 10 t = 1 Hence, from the above, We can conclude that the value of t is: 1

Question 15. 10(2y + 2) – y = 2(8y – 8)

Question 16. 2(4x + 2) = 4x – 12(x – 1) Answer: The value of x is: \(\frac{1}{2}\)

Explanation: The given equation is: 2 ( 4x + 2 ) = 4x – 12 ( x – 1 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Now, By using the Distributive Property of Multiplication, 2 ( 4x ) + 2 ( 2 ) = 4x – 12 ( x ) + 12 ( 1 ) [ Since – × – = + ] 8x + 4 = 4x – 12x + 12 8x + 4 =12 – 8x 8x + 8x = 12 – 4 16x = 8 x = 8 ÷ 16 x = \(\frac{1}{2}\) Hence, from the above, We can conclude that the value of x is: \(\frac{1}{2}\)

Question 17. MODELING WITH MATHEMATICS You and your friend drive toward each other. The equation 50h = 190 – 45h represents the number h of hours until you and your friend meet. When will you meet?

Answer: The number of movies you rent to spend the same amount at each movie store is: 20

Explanation: It is given that The equation 1.5r + 15 = 2.25r represents the number r of movies you must rent to spend the same amount at each movie store. Now, We have to find the value of r to find the number f movies you must rent to spend the same amount at each movie store So, The given equation is: 1.5r + 15 = 2.25r When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2.25r – 1.5r = 15 0.75r = 15 \(\frac{75}{100}\)r = 15 r = 15 × \(\frac{100}{75}\) r = \(\frac{15}{1}\) × \(\frac{100}{75}\) r = \(\frac{15 × 100}{1 × 75}\) r = 20 Hence, from the above, We can conclude that the number of movies you rent to spend the same amount at each movie store is: 20

In Exercises 19–24, solve the equation. Determine whether the equation has one solution, no solution, or infinitely many solutions.

Question 19. 3t + 4 = 12 + 3t

Question 20. 6d + 8 = 14 + 3d Answer: The value of d is: 2

Explanation: The given equation is: 6d + 8 = 14 + 3d When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 6d – 3d = 14 – 8 3d = 6 d = 6 ÷ 3 d = 2 Hence, from the above, We can conclude that the value of d is: 2

Question 21. 2(h + 1) = 5h – 7

Question 22. 12y + 6 = -6(2y + 1) Answer: The value of y is: –\(\frac{1}{2}\)

Explanation: The given equation is: 12y + 6 = -6 ( 2y + 1 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, By using the Distributive Property of Multiplication, 12y + 6 = -6 ( 2y ) – 6 ( 1 ) 12y + 6 = -12y – 6 12y + 12y = -6 – ( +6 ) 24y = -6 – 6 24y = -12 y = -12 ÷ 24 y = –\(\frac{1}{2}\) Hence, from the above, We can conclude that the value of y is: –\(\frac{1}{2}\)

Question 23. 3(4g + 6) = 2(6g + 9)

Question 24. 5(1 + 2m) = \(\frac{1}{2}\)(8 + 20m) Answer: m has indefinite solutions

Explanation: The given equation is: When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, By using the Distributive Property of Multiplication, 5 ( 1 ) + 5 ( 2m ) = \(\frac{1}{2}\) ( 8 ) + \(\frac{1}{2}\) ( 20m ) 2 ( 5 + 10m ) = 8 + 20m 2 ( 5 ) + 2 ( 10m ) = 8 + 20m 10 + 20m = 8 + 20m 20m – 20m = 8 – 10 20m – 20m = -2 As  m has the same coefficients and have the opposite signs, m has indefinite solutions Hence, from the above, We can conclude that the equation has the indefinite solutions

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in solving the equation.

Answer: The given equation is: 6 ( 2y + 6 ) = 4 ( 9 + 3y ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa Now, By using the Distributive Property of Multiplication, 6 ( 2y ) + 6 ( 6 ) = 4 ( 9 ) + 4 ( 3y ) 12y + 36 = 36 + 12y 12y – 12y = 36 – 36 0 = 0 As the coefficients of y are zero, the equation has no solution Hence, from the above, We can conclude that there is no error in the analysis of the equation.

Question 28. PROBLEM-SOLVING One serving of granola provides 4% of the protein you need daily. You must get the remaining 48 grams of protein from other sources. How many grams of protein do you need daily? Answer: The number of grams of protein you need daily is: 50 grams

Explanation: It is given that one serving of granola provides 4% of the protein you need daily. You must get the remaining 48 grams of protein from other sources. So, Let the number of grams of protein you need daily be: x So, The number of grams of protein you need daily = 4 % of x + 48 We know that, 100%  = 1 So, 4 % = 0.04 So, The number of grams of protein you need daily = 0.04x + 48 x = 0.04x + 48 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x – 0.04x = 48 0.96x = 48 \(\frac{96}{100}\)x = 48 x = 48 × \(\frac{100}{96}\) x = \(\frac{48}{1}\) × \(\frac{100}{96}\) x = \(\frac{48 × 100}{1 × 96}\) x = \(\frac{50}{1}\) x = 50 grams Hence, from the above, We can conclude that the number of proteins you need daily is: 50 grams

USING STRUCTURE In Exercises 29 and 30, find the value of r.

Question 29. 8(x + 6) – 10 + r = 3(x + 12) + 5x

Question 30. 4(x – 3) – r + 2x = 5(3x – 7) – 9x Answer: The value of r is: 23

Explanation: The given equation is: 4 ( x – 3 ) – r + 2x = 5 ( 3x – 7 ) – 9x When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, By using the Distributive Property of Multiplication, 4x – 4 ( 3 ) – r + 2x = 5 ( 3x ) – 5 ( 7 ) – 9x 4x – 12 – r + 2x = 15x – 35 – 9x 6x – 12 – r = 6x – 35 r = 6x – 6x – 12 + 35 r = 23 Hence, from the above, We can conclude that the value of r is: 23

MATHEMATICAL CONNECTIONS In Exercises 31 and 32, the value of the surface area of the cylinder is equal to the value of the volume of the cylinder. Find the value of x. Then find the surface area and volume of the cylinder.

Question 33. MODELING WITH MATHEMATICS A cheetah that is running 90 feet per second is 120 feet behind an antelope that is running 60 feet per second. How long will it take the cheetah to catch up to the antelope?

Question 34. MAKING AN ARGUMENT A cheetah can run at top speed for only about 20 seconds. If an antelope is too far away for a cheetah to catch it in 20 seconds, the antelope is probably safe. Your friend claims the antelope in Exercise 33 will not be safe if the cheetah starts running 650 feet behind it. Is your friend correct? Explain. Answer: Your friend is not correct

Explanation: It is given that a cheetah can run at top speed for only about 20 seconds. If an antelope is too far away for a cheetah to catch it in 20 seconds, the antelope is probably safe. Your friend claims the antelope in Exercise 33 will not be safe if the cheetah starts running 650 feet behind it. Let the distance of running antelope be x. Let ‘t’ be the time taken So, The distance of running Antelope is: x = 650 + 60t The cheetah must arrive at the same position to catch the antelope So, x = 90t Now, 90t = 650 + 60t 90t – 60t = 650 30t = 650 t = 650 ÷ 30 t = 21.7 seconds But it is given that the cheetah has to reach the same position as the antelope in 20 seconds But according to the calculation, it takes 21.7 seconds So, According to your friend, the antelope is not safe if the cheetah is running 650 meters behind it. Hence, from the above, We can conclude that your friend is not correct.

REASONING In Exercises 35 and 36, for what value of a is the equation an identity? Explain your reasoning.

Question 35. a(2x + 3) = 9x + 15 + x

Question 36. 8x – 8 + 3ax = 5ax – 2a Answer: The given equation becomes an identity at a = 4

Explanation: The given equation is: 8x – 8 + 3ax = 5ax – 2a When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5ax – 3ax = 8x – 8 – 2a 2ax = 8 ( x – 1 ) – 2a 2ax + 2a = 8x – 8 Equate the like coefficients of x and the like constants in both LHS and RHS So, 2ax = 8x                                   2a = -8 a = 8x ÷ 2x                               a = -8 ÷ 2 a = 4                                         a = -4 Now, At a = 4, The equation becomes 8x – 8 + 3ax = 5ax – 2a 8x – 8 + 3x ( 4 ) = 5x ( 4 ) -2 ( 4 ) 8x – 8 + 12x = 20x – 8 20x – 8 = 20x – 8 Hence, At a =4, The given equation is an Identity At a = -4, The equation becomes 8x – 8 + 3ax = 5ax – 2a 8x – 8 + 3x ( -4 ) = 5x ( -4 ) -2 ( -4 ) 8x – 8 – 12x = -20x + 8 -4x – 8 = -20x + 8 Hence, At a = -4, the given equation is not an Identity Hence, from the above, We can conclude that the given equation is an Identity at a = 4

Question 37. REASONING Two times the greater of two consecutive integers is 9 less than three times the lesser integer. What are the integers?

b. How does the equation 355 – 9x = 229 + 12x relate to the table and the graph? How can you use this equation to determine whether your answer in part (a) is reasonable? Answer: The given equation 355 – 9x = 229 + 12x represents the total number of students enrolled in the different years in both Spanish and French classes Now, In part (a), We observed that there is an equal enrollment of the students at 6th year in both Spanish and French classes So, Here, x is: The number of years So, In part (a), x = 6 Now, Substitute x = 6 in the given equation. Now, 355 – 9x = 229 + 12x 355 – 9 ( 6 ) = 229 + 12 ( 6 ) 355 – 54 = 229 + 72 301 = 301 As LHS = RHS We can say that the answer is reasonable in part (a)

Question 39. WRITING EQUATIONS Give an example of a linear equation that has (a) no solution and (b) infinitely many solutions. Justify your answers.

Answer: x + 3 + 2x + 1 + 3x 6x + 3 = 3(2x + 1) So, the perimeter is 3(2x + 1)

Order the values from least to greatest.

Question 41. 9, | -4|, -4, 5, | 2 |

Question 42. | -32 |, 22, -16, -| 21 |, | -10 | Answer: We know that, | -x | = x | x | = x So, | -32 | = 32 | 21 | = 21 | -10 | = 10 Hence, The order of the values from the least to the greatest is: -21, -16, 10, 22, 32

Question 43. -18, | -24 |, -19, | -18 |, | 22 |

Question 44. -| – 3 |, | 0 |, -1, | 2 |, -2 Answer: We know that, | -x | = x | x | = x So, | -3 | = 3 | 0 | = 0 | 2 | = 2 Hence, The order of the numbers from the least to the greatest is: -3, -2, -1, 0, 2

1.1-1.3 What Did You Learn

Core Vocabulary

Core Concepts

Section 1.2

Section 1.3

Mathematical Practices

Question 1. How did you make sense of the relationships between the quantities in Exercise 46 on page 9? Answer: In Exercise 46 on page 9, There is a layout of the tatami mat which comprises the four identical rectangular mats and the one square mat. it is also given that the area of the square mat is half of one of the rectangular mats Now, We know that, The area of the square mat = Area² [ Since all the sides of the square are equal ] The area of the rectangular mat = Length × Width So, According to the given condition, The relation between the area of the square mat and one of the rectangular mat is: Area of the square mat = \(\frac{1}{2}\) Area of one of the rectangular mat Side² = \(\frac{1}{2}\) ( Length × Width )

Question 2. What is the limitation of the tool you used in Exercises 25–28 on page 16? Answer: The limitations of the tool you used in Exercises 25 – 28 on page 16 are: A) The calculated values and the values measured using the tool will be different B) We won’t get the exact values of the angle measures using the tool

Question 3. What definition did you use in your reasoning in Exercises 35 and 36 on page 24? Answer: The definition you used in your reasoning in Exercises 35 and 36 on page 24 is: Make the like coefficients of the same variable in both LHS and RHS equal so that we get the value of the variable.

Study Skills

Completing Homework Efficiently

Before doing homework, review the Core Concepts and examples. Use the tutorials at BigIdeasMath.com for additional help.

Solve the equation. Justify each step. Check your solution. (Section 1.1)

Question 1. x + 9 = 7 Answer: The value of x is -2

Explanation: The given equation is: x + 9 = 7 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x = 7 –  ( +9 ) x = 7 – 9 x = -2 Hence, from the above, We can conclude that the value of x is: -2

Question 2. 8.6 = z – 3.8 Answer: The value of z is: 12.4

Explanation: The given equation is: 8.6 = z – 3.8 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, z = 8.6 + 3.8 z = 12.4 Hence, from the above, We can conclude that the value of z is: 12.4

Question 3. 60 = -12r Answer: The value of r is:  -5

Explanation: The given equation is: 60 = -12r When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa r = 60 ÷ ( -12 ) r = -60 ÷ 12 r = -5 Hence, from the above, We can conclude that the value of r is: -5

Question 4. \(\frac{3}{4}\)p = 18 Answer: The value of p is: 24

Explanation: The given equation is: \(\frac{3}{4}\)p = 18 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, p = 18 × \(\frac{4}{3}\) p = \(\frac{18}{1}\) × \(\frac{4}{3}\) p = \(\frac{18 × 4}{1 × 3}\) p = \(\frac{24}{1}\) p = 24 Hence, from the above, We can conclude that the value of p is: 24

Solve the equation. Check your solution. (Section 1.2)

Question 5. 2m – 3 = 13 Answer: The value of m is: 8

Explanation: The given equation is: 2m – 3 = 13 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2m = 13 + 3 2m = 16 m = 16 ÷ 2 m = 8 Hence, from the above, We can conclude that the value of m is: 8

Question 6. 5 = 10 – v Answer: The value of v is: 5

Explanation: The given equation is: 5 = 10 – v When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, v= 10 – 5 v = 5 Hence, from the above, We can conclude that the value of v is: 5

Question 7. 5 = 7w + 8w + 2 Answer: The value of w is: \(\frac{1}{5}\)

Explanation: The given equation is: 5 = 7w + 8w + 2 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5 – 2 = 7w + 8w 15w = 3 w = 3 ÷ 15 w = \(\frac{1}{5}\) Hence, from the above, We can conclude that the value of w is: \(\frac{1}{5}\)

Question 8. -21a + 28a – 6 = -10.2 Answer: The value of a is: –\(\frac{3}{5}\)

Explanation: The given equation is: -21a + 28a  – 6 = -10.2 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -21a + 28a = -10.2 + 6 7a = -4.2 a = -4.2 ÷ 7 a= –\(\frac{42}{10}\) ÷ 7 a = –\(\frac{42}{10}\) × \(\frac{1}{7}\) a = –\(\frac{42 × 1}{10 × 7}\) a = – \(\frac{6}{10}\) a = –\(\frac{3}{5}\) Hence, from the above, We can conclude that the value of a is: –\(\frac{3}{5}\)

Question 9. 2k – 3(2k – 3) = 45 Answer: The value of k is: -9

Explanation: The given equation is: 2k – 3 ( 2k – 3 ) = 45 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa SO, 2k – 3 ( 2k ) + 3 ( 3 ) = 45 2k – 6k + 9 = 45 2k – 6k = 45 – 9 -4k = 36 k = 36 ÷ -4 k = -9 Hence, from the above, We can conclude that the value of k is: -9

Question 10. 68 = \(\frac{1}{5}\)(20x + 50) + 2 Answer: The value of x is: 14

Explanation: The given equation is: 68 = \(\frac{1}{5}\) [ 20x + 50 ] + 2 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 68 – 2 = \(\frac{1}{5}\) [ 20x + 50 ] 66 = \(\frac{1}{5}\) [ 20x + 50 ] 66 × 5 = 20x + 50 330 = 20x + 50 20x = 330 – 50 20x = 280 x = 280 ÷ 20 x = 14 Hence, from the above, We can conclude that the value of x is: 14

Solve the equation. (Section 1.3)

Question 11. 3c + 1 = c + 1 Answer: The value of c is: 0

Explanation: The given equation is: 3c + 1 = c + 1 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa 3c – c = 1 – 1 2c = 0 c = 0 Hence, from the above, We can conclude that the value of c is: 0

Question 12. -8 – 5n = 64 + 3n Answer: The value of n is: -9

Explanation: The given equation is: -8 – 5n = 64 + 3n When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -8 – 64 = 3n + 5n -72 = 8n n = -72 ÷ 8 n = -9 Hence, from the above, We can conclude that the value of n is: -9

Question 13. 2(8q – 5) = 4q Answer: The value of q is: \(\frac{5}{6}\)

Explanation: Te given equation is: 2 ( 8q – 5 ) = 4q When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2 ( 8q ) – 2 ( 5 ) = 4q 16q – 10 = 4q 16q – 4q = 10 12q = 10 q = 10 ÷ 12 q = \(\frac{5}{6}\) Hence, from the above, We can conclude that the value of q is: \(\frac{5}{6}\)

Question 14. 9(y – 4) – 7y = 5(3y – 2) Answer: The value of y is: -2

Explanation: The given equation is: 9 ( y – 4 ) – 7y = 5 ( 3y – 2 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 9 ( y ) – 9 ( 4 ) – 7y = 5 ( 3y ) – 5 ( 2 ) 9y – 36 – 7y = 15y – 10 2y – 36 = 15y – 10 15y – 2y = 10 – 36 13y = -26 y = -26 ÷ 13 y = -2 Hence, from the above, We can conclude that the value of y is: -2

Question 15. 4(g + 8) = 7 + 4g Answer: The given equation has no solution

Explanation: The given equation is: 4 ( g + 8 ) = 7 + 4g When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So,the given equation has no solution + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4 ( g ) + 4 ( 8 ) = 7 + 4g 4g + 32 = 7 + 4g 4g – 4g + 7 = 32 7 = 32 Hence, from the above, We can conclude that the given equation has no solution.

Question 16. -4(-5h – 4) = 2(10h + 8) Answer: The value of h is: 0

Explanation: The given equation is: -4 ( 5h – 4 ) = 2 ( 10h + 8 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -4 ( 5h ) + 4 ( 4 ) = 2 ( 10h ) + 2 ( 8 ) -20h + 16 = 20h + 16 -20h – 20h = 1 – 16 -40h =0 h = 0 Hence, from the above, We can conclude that the value of h is: 0

Question 17. To estimate how many miles you are from a thunderstorm, count the seconds between when you see lightning and when you hear thunder. Then divide by 5. Write and solve an equation to determine how many seconds you would count for a thunderstorm that is 2 miles away. (Section 1.1) Answer: Given, To estimate how many miles you are from a thunderstorm, count the seconds between when you see lightning and when you hear thunder. Then divide by 5. x/5 = 2 x = 2 × 5 x = 10

Answer: The space you should leave between the posters is: \(\frac{3}{2}\) ft

Answer: a) It is given that, The total cost = Cost of the vase + The hourly studio fee Let the number of hours be x So, The total cost for studio A = 10 + 8x The total cost of studio B = 16 + 6x It is given that the total costs are the same So, 10 + 8x = 16 + 6x 8x – 6x = 16 – 10 2x = 6 x = 6 ÷ 2 x = 3 Hence, from the above, We can conclude that the total cost will be the same after 3 hours for both the studios

b) It is given that the studio B increases the hourly studio fee by $2 So, The total hourly studio fee for studio B = 6 + 2 = $8 So, Now, As in part (a), the same process will be repeated but in the studio B’s hourly fee of $6, we have to put $8 So, 10 + 8x = 16 + 8x 8x – 8x = 6 – 10 10 = 16 Hence, from the above, We can conclude that the value of x has no solutions

b. Use the linear equations you wrote in part (a) to find the solutions of the absolute value equation. Answer: We know that, | x | = x -| x | = -x So, | x + 2 | = 3 x + 2 = 3 x = 3 – 2 x = 1 Now, -| x + 2 | = 3 -| x + 2  | = -3 -x – 2 = -3 -x = -3 + 2 -x = -1 x = 1 So, The solutions of | x + 2 | are: 1 and 1

c. How can you use linear equations to solve an absolute value equation? Answer: We use linear equations to solve an absolute value equation by using the following properties. They are: A) | x | = x B) -| x | = -x

Question 4. How can you solve an absolute value equation? Answer: We can solve the absolute equation by using the following properties. They are: A) | x| = x B) – | x | = -x

Question 5. What do you like or dislike about the algebraic, graphical, and numerical methods for solving an absolute value equation? Give reasons for your answers. Answer: The algebraic, numerical, and graphical methods have their own advantages in their own perspective. The algebraic methods used to solve the linear equations whereas the graphical method used to indicate the linear equations. The numerical method is applicable for mathematical operations

Solve the equation. Graph the solutions, if possible.

Question 1. | x | = 10 Answer: The value of x is: 10

Explanation: The given absolute value equation is: | x | = 10 We know that, | x | = x So, x = 10 Hence, from the above, We can conclude that the value of x is: 10

Question 2. | x – 1 | = 4 Answer: The value of x is: 5

Explanation: The given absolute value equation is: | x – 1 | = 4 We know that, | x | = x So, x – 1 = 4 x = 4 + 1 x = 5 Hence, from the above, We can conclude that the value of x is: 5

Question 3. | 3 + x | = -3 Answer: The value of x is: -6

Explanation: The given absolute value equation is: | 3 + x | = -3 We know that, | x | = x So, x + 3 = -3 x = -3 – 3 x = -6 Hence, from the above, We can conclude that the value of x is: -6

Solve the equation. Check your solutions.

Question 4. | x – 2 | + 5 = 9 Answer: The value of x is: 6

Explanation: The given absolute value equation is: | x – 2 | + 5 = 9 | x – 2 | = 9 – 5 | x – 2 | = 4 We know that, | x | = x So, x – 2 = 4 x = 4 + 2 x = 6 Hence, from the above, We can conclude that the value of x is: 6

Question 5. 4 | 2x + 7 | = 16 Answer: The value of x is: –\(\frac{3}{2}\)

Explanation: The given absolute value equation is: 4 | 2x + 7 | = 16 | 2x + 7 | = 16 ÷ 4 | 2x + 7| = 4 We know that, | x | = x So, 2x + 7 = 4 2x = 4 – 7 2x = -3 x = –\(\frac{3}{2}\) Hence, from the above, We can conclude that the value of x is: –\(\frac{3}{2}\)

Question 6. -2 | 5x – 1 | – 3 = -11 Answer: The value of x is: 1

Explanation: The given absolute value equation is: -2 | 5x – 1 | – 3 = -11 -2 | 5x – 1 | = -11 + 3 -2 | 5x – 1 | = -8 | 5x – 1 | = -8 ÷ ( -2 ) | 5x – 1 | = 4 We know that, | x | = x So, 5x – 1 = 4 5x = 4 + 1 5x = 5 x = 5 ÷ 5 x = 1 Hence, from the above, We can conclude that the value of x is: 1

Question 7. For a poetry contest, the minimum length of a poem is 16 lines. The maximum length is 32 lines. Write an absolute value equation that represents the minimum and maximum lengths. Answer: The minimum value length is: 16 The maximum length is: 32

Explanation: It is given that for a poetry contest, the minimum length of a poem is 16 lines. The maximum length is 32 lines. So, The absolute value equation that represents the minimum length of a poem = | The minimum length of a poem | = | 16 | = 16 The absolute value equation that represents the maximum length of a poem = | The maximum length of a poem | = | 32 | = 32 Hence, from the above, We can conclude that The minimum value length is: 16 The maximum length is: 32

Question 8. | x + 8 | = | 2x + 1 | Answer: The value of x is: 7

Explanation: The given absolute value equation is: | x + 8 | = | 2x + 1 | We know that, | x | = x So, x + 8 = 2x + 1 2x – x = 8 – 1 x = 7 Hence from the above, We can conclude that the value of x is: 7

Question 9. 3 | x – 4 | = | 2x + 5 | Answer: The value of x is: 17

Explanation: The given absolute equation is: 3 | x – 4 | = | 2x + 5 | We know that, | x |  = x So, 3 ( x – 4 ) = 2x + 5 3 ( x ) – 3 ( 4 ) = 2x + 5 3x – 12 = 2x + 5 3x – 2x = 5 + 12 x = 17 Hence, from the above, We can conclude that the value of x is: 17

Question 10. | x + 6 | = 2x Answer: The value of x is: 6

Explanation: The absolute value equation is: | x + 6 | = 2x We know that, | x | = x So, x + 6 = 2x 2x – x = 6 x = 6 Hence, from the above, We can conclude that the value of x is: 6

Question 11. | 3x – 2 | = x Answer: The value of x is: 1

Explanation: The given absolute value equation is: | 3x – 2 | = x We know that, | x | = x So, 3x – 2 = x Soo, 3x – x = 2 2x = 2 x = 2 ÷ 2 x = 1 Hence, from the above, We can conclude that the value of x is: 1

Question 12. | 2 + x | = | x – 8 | Answer: The given absolute value equation has no solution

Explanation: The given absolute value equation is: | 2 + x | = | x – 8 | We know that, | x | = x So, 2 + x = x – 8 2 = x – x – 8 2 = -8 Hence, from the above, We can conclude that the given absolute value equation has no solution

Question 13. | 5x – 2 | = | 5x + 4 | Answer: The given absolute value equation has no solution

Explanation: The given absolute value equation is: | 5x – 2 | = | 5x + 4 | We know that, | x | = x So, 5x – 2 = 5x + 4 5x – 5x – 2 = 4 -2 = 4 Hence, from the above, We can conclude that the given absolute value equation has no solution

Vocabulary and Core Concept Check

Question 1. VOCABULARY What is an extraneous solution? Answer: Extraneous solutions are values that we get when solving equations that are not really solutions to the equation. Example for extraneous solution: | 5x – 2 | = | 5x + 4 |

Question 2. WRITING Without calculating, how do you know that the equation | 4x – 7 | = -1 has no solution? Answer: The given absolute value equation is: | 4x – 7 | = -1 We know that, An absolute value can never equal a negative number. So, By the above, We can say that | 4x – 7 | must not equal to a negative number. Hence, from the above, We can conclude that | 4x – 7 | = -1 has no solution without calculating its solution

In Exercises 3−10, simplify the expression.

Question 3. | -9 |

Question 4. – | 15 | Answer: The value of -| 15 | is: -15

Explanation: The given absolute value is: -| 15 | We know that, | x | = x | -x | = x -| x | = -x So, -| 15 | = -15 Hence, from the above, We can conclude that the value of -| 15 | is: -15

Question 5. | 14 | – | -14 |

Question 6. | -3 | + | 3 | Answer:

The value of | -3 | + | 3 | is: 6

Explanation: The given absolute value expression is: | -3 | + | 3 | We know that, | x | = x | -x | = x -| x | = -x So, | -3 | + | 3 | = 3 + 3 = 6 Hence, from the above, We can conclude that the value of | -3 | + | 3 | is: 6

Question 7. – | -5 • (-7) |

Question 8. | -0.8 • 10 | Answer: The value of | -0.8 ⋅ 10 | is: 8

Explanation: The given absolute value expression is: | -0.8 ⋅ 10 | We know that, | x | = x | -x | = x -| x | = -x So, | -0.8 ⋅ 10 | = | – ( 8 ⁄ 10 ) ⋅ ( 10 ⁄ 1 ) | = | – ( 8 × 10 ) ⁄ ( 10 × 1 ) | = | -8 | = 8 hence, from the above, We can conclude that the value of | -0.8 ⋅ 10 | is: 8

Explanation: The given absolute value expression is: | -12 ⁄ 4 | We know that, | x | = x | -x | = x – | x | = -x So, | -12 ⁄ 4 | = | -3 | = 3 Hence, from the above, We can conclude that the value of | -12 ⁄ 4 | is: 3

In Exercises 11−24, solve the equation. Graph the solution(s), if possible.

Question 11. | w | = 6

Question 12. | r | = -2 Answer: The absolute value of a number must be greater than or equal to 0 and can not be equal to -2. Hence, The given absolute eqution has no solution

Question 13. | y | = -18

Question 14. | x | = 13 Answer: The value of x is: 13 or -13

Explanation: The given absolute value equation is: | x | = 13 We know that, |x | = x – | x | = -x So, | x | = 13 or – 13 Hence, from the above, We can conclude that the value of x is: 13 or -13

Question 15. | m + 3 | = 7

Question 16. | q – 8 | = 14 Answer: The value of q is: 22 or -6

Explanation: The given absolute value equation is: | q – 8 | = 14 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, q – 8 = 14                                                      q – 8 = -14 q = 14 + 8                                                     q = -14 + 8 q = 22                                                             q = -6 Hence, from the above, We can conclude that the value of q is: 22 or -6

Question 17. | -3d | = 15

Explanation: The given absolute value equation is: | t / 2 | = 12 We know that, | x | = x  for x > 0 | x | = -x for x < 0 So, t / 2 = 6                                       t / 2 = -6 t = 6 × 2                                      t = 6 × -2 t = 12                                           t = -12 Hence, from the above, We can conclude that the value of t is: 12 or -12

Question 19. | 4b – 5 | = 19

Question 20. | x – 1 | + 5 = 2 Answer: The given absolute value equation has no solution

Explanation: The given absolute value equation is: | x – 1 | + 5 = 2 | x – 1 | = 2 – 5 | x – 1  | = -3 We know that, The absolute value of an equation must be greater than or equal to zero So, | x – 1 | = -3 has no solution Hence, from the above, We can conclude that the given absolute value equation has no solution

Question 21. -4 | 8 – 5n | = 13

Explanation: The given absolute value equation is: -3 | 1 – ( 2 / 3 ) y | = -9 | 1 – (2 / 3  ) y | = -9 ÷ ( -3 ) | 1 – ( 2 / 3 ) y | = 3 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 1 – ( 2 / 3 ) y = 3                                    1 – ( 2 /3 ) y = -3 2/3 y = 1 – 3                                           2/3 y = 1 + 3 2 / 3 y = -2                                              2 / 3 y = 4 2y = -2 × 3                                               2y = 4 × 3 2y = -6                                                      2y = 12 y = -6 ÷ 2                                                  y = 12 ÷ 2 y = -3                                                         y = 6 Hence, from the above, We can conclude that the value of y is: -3 or 6

Question 24. 9 | 4p + 2 | + 8 = 35 Answer: The value of p is: 1 / 4 or -5 / 4

Explanation: The given absolute value equation is: 9 | 4p + 2 | + 8 = 35 9 | 4p + 2 | = 35 – 8 9 | 4p + 2 | = 27 | 4p + 2 | = 27 ÷ 9 | 4p + 2 | = 3 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 4p + 2 = 3                    4p + 2 = -3 4p = 3 – 2                     4p = -3 – 2 4p = 1                           4p = -5 p = 1 / 4                         p = -5 / 4 Hence, from the above, We can conclude that the value of p is: 1 / 4 or -5 / 4

Question 25. WRITING EQUATIONS The minimum distance from Earth to the Sun is 91.4 million miles. The maximum distance is 94.5 million miles. a. Represent these two distances on a number line. b. Write an absolute value equation that represents the minimum and maximum distances.

USING STRUCTURE In Exercises 27−30, match the absolute value equation with its graph without solving the equation.

Question 27. | x + 2 | = 4

Question 28. | x + 4 | = 2 Answer: The given absolute value equation is: | x + 4 | = 2 To find the halfway point, made the absolute value equation equal to 0. So, | x + 4  | = 0 So, x = -4 From the given absolute value equation, We can say that the distance from the halfway point to the minimum and maximum points is: 2

Question 29. | x – 2 | = 4

Answer: A is correct

In Exercises 31−34, write an absolute value equation that has the given solutions.

Question 31. x = 8 and x = 18

Question 32. x = -6 and x = 10 Answer: The given absolute value equation is: | x – 2 | = 8

Explanation: The given values of x are: x = -6 and x = 10 Now, The halfway point between 10 and -6 = [ 10 – ( -6 ) ] / 2 = [ 10 + 6 ] / 2 = 16 / 2 = 8 The minimum distance from the halfway point = 8 – 6 = 2 Hence, The absolute value equation is: | x – 2 | = 5

Question 33. x = 2 and x = 9

Question 34. x = -10 and x = -5

Answer: The given values of x are: x = -10 and x = -5 Now, The halfway point between -10 and -5 = [ 10 – ( 5 ) ] / 2 = [ 10 – 5 ] / 2 = 5 / 2 = 2.5 So, The minimum value from the half-point = 2.5 + ( -10 ) = 2.5 – 10 = -7.5 Hence, The absolute value equation is: | x – ( -7.5 ) | = 2.5 | x + 7.5 | = 2.5

In Exercises 35−44, solve the equation. Check your solutions. 

Question 35. | 4n – 15 | = | n |

Question 36. | 2c + 8 | = | 10c | Answer: The values of c are: 1 and 2 / 3

Explanation: The given absolute value equation is: | 2c + 8 | = | 10c | We know that, | x | = x for x > 0 | x | = -x for x < 0 Now, 2c + 8 = 10c                                      2c + 8 = -10c 10c – 2c = 8                                       2c + 10c = 8 8c = 8                                                12c = 8 c = 8 / 8                                              c = 8 / 12 c = 1                                                    c = 2 /3 Hence, from the above, We can conclude that the values of c are: 1 and 2 / 3

Question 37. | 2b – 9 | = | b – 6 |

Question 38. | 3k – 2 | = 2 | k + 2 | Answer: The values of k are: 6 and -2 / 5

Explanation: The given absolute equation is: | 3k – 2 | = 2 | k + 2 | We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 2 ( k + 2 ) = 3k – 2                                            2 ( k + 2 ) = – ( 3k – 2 ) 2k + 4 = 3k – 2                                                 2k + 4 = -3k + 2 3k – 2k = 4 + 2                                                 2k + 3k = 2 – 4 k = 6                                                                 5k = -2 k = 6                                                                 k = -2 / 5 Hence, from the above, We can conclude that the values of k are: 6 and -2 / 5

Question 39. 4 | p – 3 | = | 2p + 8 |

Question 40. 2 | 4w – 1 | = 3 | 4w + 2 | Answer: The value of w is: -2

Explanation: The given absolute value equation is: 2 | 4w – 1 | = 3 | 4w+ 2 | We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 2 ( 4w – 1 ) = 3 ( 4w + 2 )                        -2 ( 4w – 1 ) = -3 ( 4w + 2 ) 8w – 2 = 12w + 6                                      -8w + 2 = -12w -6 12w – 8w = -6 – 2                                      -12w + 8w = 6 + 2 4w = -8                                                       -4w = 8 w = -8 / 4                                                      w = 8 / -4 w = -2                                                            w = -2 Hence, from the above, We can conclude that the value of w is: -2

Question 41. | 3h + 1 | = 7h

Question 42. | 6a – 5 | = 4a Answer: The value of a is: 5 / 2 and 1 / 2

Explanation: The given absolute value equation is: | 6a – 5 | = 4a We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 6a – 5 = 4a                                                6a – 5 = -4a 6a – 4a = 5                                                 6a + 4a = 5 2a = 5                                                         10a = 5 a = 5 / 2                                                       a = 5 / 10 a = 5 / 2                                                       a = 1 / 2 Hence, from the above, We can conclude that the values of a are: 5 / 2 and 1 / 2

Question 43. | f – 6 | = | f + 8 |

Question 44. | 3x – 4 | = | 3x – 5 | Answer: The given absolute value equation has no solution

Explanation: The given absolute value equation is: | 3x – 4 | = | 3x – 5 | We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 3x – 4 = 3x – 5                                – ( 3x – 4 ) = – ( 3x – 5 ) 4 = 5                                                 4 = 5 Hence, from the above, We can conclude that the given absolute value equation has no solution

Question 45. MODELING WITH MATHEMATICS Starting from 300 feet away, a car drives toward you. It then passes by you at a speed of 48 feet per second. The distance d (in feet) of the car from you after t seconds is given by the equation d = | 300 – 48t |. At what times is the car 60 feet from you?

Question 46. MAKING AN ARGUMENT Your friend says that the absolute value equation | 3x + 8 | – 9 = -5 has no solution because the constant on the right side of the equation is negative. Is your friend correct? Explain. Answer: Yes, your friend is correct

Explanation: The given absolute value equation is: | 3x + 8 | – 9 = -5 We know that, The absolute value equation value must have greater than or equal to 0 But here The value of the absolute value equation is less than 0 Hence, The given absolute value equation has no solution. Hence, from the above, We can conclude that your friend is correct.

Question 48. MODELING WITH MATHEMATICS The recommended weight of a soccer ball is 430 grams. The actual weight is allowed to vary by up to 20 grams. a. Write and solve an absolute value equation to find the minimum and maximum acceptable soccer ball weights. Answer: It is given that the recommended weight of a soccer ball is 430 grams and the actual weight is allowed to vary up to 20 grams Hence, The absolute value equation that represents the minimum and maximum acceptable soccer ball weights is: | x – 430 | = 20 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, x – 430 = 20                               x – 430 = -20 x = 20 + 430                              x = -20 + 430 x = 460 grams                            x = 410 grams Hence, from the above, We can conclude that the maximum and minimum acceptable soccer weights respectively are: 460 grams and 410 grams

b. A soccer ball weighs 423 grams. Due to wear and tear, the weight of the ball decreases by 16 grams. Is the weight acceptable? Explain. Answer: The weight that caused due to wear and tear is not acceptable

Explanation: From the above problem, We get the maximum weight of the soccer ball to be 460 grams with 20 grams increase or decreased to the weight of the ball Now, It is given that the weight of the ball is decreased by 16 grams due to wear and tear So, The weight of the ball now = 460 – 16 = 444 grams But it is given that the weight of the ball becomes 423 grams due to wear and tear. Hence, from the above, We can conclude that the weight is not acceptable

ERROR ANALYSIS In Exercises 49 and 50, describe and correct the error in solving the equation.

Answer: The values of x are: -2 and -4 / 3

Explanation: The given absolute value equation is: | 5x + 8 | = x We know that, | x | = x for x > 0 | x | = – x for x < 0 So, 5x + 8 = x                                                      5x + 8 = -x 5x – x = -8                                                      5x + x = -8 4x = -8                                                            6x = -8 x = -8 / 4                                                         x = -8 / 6 x = -2                                                               x = -4 / 3 Hence, from the above, We can conclude that the values of x are: -2 and -4 / 3

ABSTRACT REASONING In Exercises 53−56, complete the statement with always, sometimes, or never. Explain your reasoning.

Question 53. If x 2 = a 2 , then | x | is ________ equal to | a |.

Question 54. If a and b are real numbers, then | a – b | is _________ equal to | b – a |. Answer: If a and b are real numbers, then | a – b | is equal to | b – a |

Explanation: Let, | a | = 5 and | b | = 9 We know that, | x | =x for  x > 0 | x | = -x for x < 0 So, | a – b | = | 5 – 9 | = | -4 | = 4 | b – a | = | 9 – 5 | =  | 4 | = 4 Hence, from the above, We can conclude that value of | a – b | is equal to | b – a | if a and b are real numbers

Question 55. For any real number p, the equation | x – 4 | = p will ________ have two solutions.

Question 56. For any real number p, the equation | x – p | = 4 will ________ have two solutions. Answer: For any real number, | x – p | = 4 will have two solutions

Explanation: The given absolute value equation is: | x – p | = 4 Let the value of p be 1 We know that, | x | = x for x > 0 | x | = – x for x < 0 So, | x – 1 | = 4 | x – 1 | = 4                                        | x – 1 | = -4 x = 4 + 1                                           x = -4 + 1 x = 5                                                  x = -3 Hence, from the above, We can conclude that | x – p | = 4 will have two solutions for any real number p

Question 57. WRITING Explain why absolute value equations can have no solution, one solution, or two solutions. Give an example of each case.

Question 58. THOUGHT-PROVOKING Describe a real-life situation that can be modeled by an absolute value equation with the solutions x = 62 and x = 72. Answer: Suppose in a school, an exam is conducted. In that examination, 67% of the students are passed. If the error of the pass percentage is 5 %, then what are the minimum and the maximum number of students passed in the examination? Now, The absolute value equation for the given real-life situation is: | x – 67 | = 5 We know that, | x | = x for x> 0 | x | =-x for x < 0 So, x – 67 = 5                                         x – 67 = -5 x = 5 + 67                                        x = -5 + 67 x = 72                                               x = 62 Hence, from the above, We can conclude that the minimum and maximum number of students passed in the examination  respectively are: 72 and 67

Question 59. CRITICAL THINKING Solve the equation shown. Explain how you found your solution(s). 8 | x + 2 | – 6 = 5 | x + 2 | + 3

b. How can you use absolute value equations to represent your answers in part (a)? Answer: From the property of absolute values, We know that, | x | = x for x > 0 | x | = -x for x < 0 From the part ( a ), The absolute value equation is: | x – 42 | = 2 So, x – 42 = 2                                                                 x – 42 = -2 x = 2 + 42                                                                x = -2 + 42 x = 44                                                                       x = 40 Hence, from the above, We can conclude that we can use absolute values in the above way to represent the answers

c. One candidate receives 44% of the vote. Which party does the candidate belong to? Explain. Answer: The candidate of the Republican party receives 44 % of the vote.

Question 61. ABSTRACT REASONING How many solutions does the equation a | x + b | + c = d have when a > 0 and c = d? when a < 0 and c > d? Explain your reasoning.

Identify the property of equality that makes Equation 1 and Equation 2 equivalent. (Section 1.1)

Answer: The given equations are: Equation 1: 3x + 8 = x – 1 Equation 2: 3x + 9= x From Equation 1, 3x + 8 = x – 1 3x + 8 + 1 = x 3x + 9 = x Hence, from the above, We can conclude that we can get Equation2 by rearranging the Equation 1

Use a geometric formula to solve the problem.

Question 64. A square has an area of 81 square meters. Find the side length. Answer: The side length of the square is: 9 meters

Explanation: The given area of the square is: 81 square meters We know that, Area of the square = Side × Side 81 = Side × Side Side² = 81 Apply square root on both sides √Side² = √81 Side = 9 meters Hence, from the above, We can conclude that the side of the square is: 9 meters

Question 65. A circle has an area of 36π square inches. Find the radius.

Question 66. A triangle has a height of 8 feet and an area of 48 square feet. Find the base. Answer: The base of the triangle is: 12 feet

Explanation: It is given that a triangle has a height of 8 feet and an area of 48 square feet We know that, The area of the triangle = ( 1 /  2 ) × Base × Height 48 = ( 1 / 2 ) × Base × 8 Base × 8 = 48 × 2 Base = ( 48 × 2 ) ÷ 8 Base = 96 ÷ 8 Base = 12 feet Hence, from the above, We can conclude that the base of the triangle is: 12 feet

Question 67. A rectangle has a width of 4 centimeters and a perimeter of 26 centimeters. Find the length.

Essential Question How can you use a formula for one measurement to write a formula for a different measurement? Answer: Write the formula for one measurement and then solve the formula for the different measurement you want to find and use this new formula to find that measurement Hence, in the above way, We can use a formula for one measurement to write a formula for a different measurement

EXPLORATION 1 Using an Area Formula Work with a partner. a. Write a formula for the area A of a parallelogram. Answer: We know that, The area of the parallelogram ( A) = Base × Height

c. Solve the formula in part (a) for b without first substituting values into the formula. Justify each step. Answer: From part ( a ), Area of the parallelogram = Base × Height Base = ( Area of the parallelogram ) ÷ Height of the parallelogram From the given figure, Base = b So, b = ( Area of the parallelogram ) ÷ Height of the parallelogram

Solve the literal equation for y.

Question 1. 3y – x = 9 Answer: The value of y is: ( x + 9 ) / 3

Explanation: The given equation is: 3y – x = 9 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 3y = 9 + x y = ( x + 9 ) / 3 Hence, from the above, We can conclude that the value of y is: ( x + 9 ) / 3

Question 2. 2x – 2y = 5 Answer: The value of y is: ( 2x – 5 ) / 2

Explanation: The given equation is: 2x – 2y = 5 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 2y = 2x – 5 y = ( 2x – 5 ) / 2 Hence, from the above, We can conclude that the value of y is: ( 2x – 5 ) / 2

Question 3. 20 = 8x + 4y Answer: The value of y is: 5 – 2x

Explanation: The given equation is: 20 = 8x + 4y When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4y = 20 – 8x y = ( 20 – 8x ) / 4 y = ( 20 ÷ 4 ) – ( 8x ÷ 4 ) y = 5 – 2x Hence, from the above, We can conclude that the value of y is: 5 – 2x

Solve the literal equation for x.

Question 4. y = 5x – 4x Answer: The value of x is: y

Explanation: The given equation is: y = 5x – 4x When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x = y Hence, from the above, We can conclude that the value of x is: y

Question 5. 2x + kx = m Answer: The value of x is: m / ( k + 2  )

Explanation: The given equation is: 2x + kx = m When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, x ( k + 2 ) = m x = m / ( k + 2 ) Hence, from the above, We can conclude that the value of x is: m / ( k + 2 )

Question 6. 3 + 5x – kx = y Answer: The value of x is: ( y – 3 ) / ( 5 – k )

Explanation: The given equation is: 3 + 5x – kx = y When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5x – kx = y – 3 x ( 5 – k ) = y – 3 x = ( y – 3 ) / ( 5 – k ) Hence, from the above, We can conclude that the value of x is: ( y – 3 ) / ( 5 – k )

Solve the formula for the indicated variable. 

Question 7. Area of a triangle: A = \(\frac{1}{2}\)bh; Solve for h. Answer: The value of h is: \(\frac{2A}{b}\)

Explanation: The given area of a triangle is: A = \(\frac{1}{2}\) bh bh = 2A h = \(\frac{2A}{b}\) Hence, from the above, We can onclude that the value of h is: \(\frac{2A}{b}\)

Question 8. The surface area of a cone: S = πr 2 + πrℓ; Solve for ℓ. Answer: The value of l is: \(\frac{S}{πr}\) – r

Explanation: The given surface area of a cone is: S = πr² + πrl S= πr ( r + l ) r + l = \(\frac{S}{πr}\) l = \(\frac{S}{πr}\) – r Hence, from the above, We can conclude that the value of l is: \(\frac{S}{πr}\) – r

Question 9. A fever is generally considered to be a body temperature greater than 100°F. Your friend has a temperature of 37°C. Does your friend have a fever? Answer: Your friend does not have a fever

Explanation: It is given that a fever is generally considered to be a body temperature greater than 100°F. We know that, To convert Fahrenheit into Celsius, °C = ( °F – 32 ) × \(\frac{5}{9}\) °C = ( 100 – 32 ) × \(\frac{5}{9}\) °C = 68 × \(\frac{5}{9}\) °C = 37.7° But it is given that your friend has a temperature of 37°C So, for fever, the temperature has to be 37.7°C Hence, from the above, We can conclude that your friend does not have a fever

Question 10. How much money must you deposit in a simple interest account to earn $500 in interest in 5 years at 4% annual interest? Answer: The money you deposit in simple interest is: $2,500

Explanation: It is given that you earned $500 in a simple interest to earn in 5 years at 4% annual interest Let, The money you deposited be: $x We know that, Simple interest = ( Principle × Time × Rate ) / 100 The principle is the money you deposited So, 500 = ( x × 5 × 4 ) / 100 ( x × 5 × 4 ) = 500 × 100 x × 20 = 500 × 100 x = ( 500 × 100 ) ÷ 20 x = $2,500 Hence, from the above, We can conclude that the money you deposited is: $2,500

Question 11. A truck driver averages 60 miles per hour while delivering freight and 45 miles per hour on the return trip. The total driving time is 7 hours. How long does each trip take? Answer: The time taken for each trip is: 3 hours and 4 hours respectively

Explanation: It is given that a truck driver averages 60 miles per hour while delivering freight and 45 miles per hour on the return trip. The total driving time is 7 hours. We know that, Speed = \(\frac{Distance}{Time}\) Time = \(\frac{Distance}{Speed}\) Let the distance be D It is given that the total driving time is: 7 hours So, 7 = \(\frac{D}{60}\) + \(\frac{D}{45}\) 7 / D = \(\frac{60 + 45}{60 × 45}\) 7 / D = \(\frac{105}{2,700}\) D = 7 / \(\frac{105}{2,700}\) D = 7 × \(\frac{2,700}{105}\) D = \(\frac{7}{1}\) × \(\frac{2,700}{105}\) D = \(\frac{7 × 2,700}{1 × 105}\) D = 180 miles So, The time taken to deliver = \(\frac{180}{60}\) = 3 hours The time taken to return = \(\frac{180}{45}\) = 4 hours Hence, from the above, We can conclude that the time taken for each trip is: 3 hours and 4 hours respectively

Rewriting Equations and Formulas 1.5 Exercices

In Exercises 3–12, solve the literal equation for y.

Question 3. y – 3x = 13

Question 4. 2x + y = 7 Answer: The value of y is: 7 – 2x

Explanation: The given literal equation is: 2x + y = 7 Now, y = 7 – 2x Hence, from the above, We can conclude that the value of y is: 7 – 2x

Question 5. 2y – 18x = -26

Question 6. 20x + 5y = 15 Answer: The value of y is: 3 – 4x

Explanation: The given literal equation is: 20x + 5y = 15 Now, 5y = 15 – 20x y = ( 15 – 20x ) / 5 y = ( 15  / 5 ) – ( 20x / 5 ) y = 3 – 4x Hence, from the above, We can conclude that the value of y is: 3 – 4x

Question 7. 9x – y = 45

Question 8. 6 – 3y = -6 Answer: The value of y is: 4

Explanation: The given literal equation is: 6 – 3y = -6 -3y = -6 – ( +6 ) -3y = -6 -6 -3y = -12 y = -12 ÷ ( -3 ) y = 12 ÷ 3 y = 4 Hence, from the above, We can conclude that the value of y is: 4

Question 9. 4x – 5 = 7 + 4y

Question 10. 16x + 9 = 9y – 2x Answer: The value of y is: 18x + 9

Explanation: The given literal equation is: 16x + 9 = y – 2x So, 16x + 2x + 9 = y 18x + 9 = y y = 18x + 9 Hence, from the above, We can conclude that the value of y is: 18x + 9

Question 11. 2 +\(\frac{1}{6}\)y = 3x + 4

Question 12. 11 – \(\frac{1}{2}\)y = 3 + 6x Answer: The value of y is: 16 – 12x

Explanation: The given literal equation is: 11 – \(\frac{1}{2}\)y = 3 + 6x So, –\(\frac{1}{2}\)y = 3 + 6x – 11 -y = 2 ( 3 + 6x – 11 ) y = -2 ( 3 + 6x – 11 ) y = -2 ( 3 ) -2 ( 6x ) + 2 ( 11 ) y = -6 – 12x + 22 y = 16 – 12x Hence, from the above, We can conclude that the value of y is: 16 – 12x

In Exercises 13–22, solve the literal equation for x.

Question 13. y = 4x + 8x

Question 14. m = 10x – x Answer: The value of x is: m / 9

Explanation: The given literal equation is: m = 10x – x m = 9x x = m / 9 Hence, from the above, We can conclude that the value of x is: m / 9

Question 15. a = 2x + 6xz

Question 16. y = 3bx – 7x Answer: The value of x is: y / ( 3b – 7 )

Explanation: The given literal equation is: y = 3bx – 7x So, y = x ( 3b – 7 ) x = y / ( 3b – 7 ) Hence, from the above, We can conclude that the value of x is: y / ( 3b – 7 )

Question 17. y = 4x + rx + 6

Question 18. z = 8 + 6x – px Answer: The value of x is: ( z – 8 ) / ( 6 – p )

Explanation: The given literal equation is: z = 8 + 6x – px So, z – 8 = 6x – px z – 8 = x ( 6 – p ) x = ( z – 8 ) / ( 6 – p ) Hence, from the above, We can conclude that the value of x is: ( z – 8 ) / ( 6 – p )

Question 19. sx + tx = r

Question 20. a = bx + cx + d Answer: The value of x is: ( a – d ) / ( b + c )

Explanation: The given literal equation is: a = bx + cx + d a – d = bx + cx a – d = x ( b + c ) x = ( a – d ) / ( b + c ) Hence, from the above, We can conclude that the value of x is: ( a – d ) / ( b + c )

Question 21. 12 – 5x – 4kx = y

Question 22. x – 9 + 2wx = y Answer: The value of x is: ( y – 9 ) / ( 1 – 2w )

Explanation: The given literal equation is: x – 9 + 2wx = y x – 2wx = y + 9 x ( 1 – 2w ) = y + 9 x = ( y – 9 ) / ( 1 – 2w ) Hence, from the above, We can conclude that the value of x is: ( y – 9 ) / ( 1 – 2w )

Question 24. MODELING WITH MATHEMATICS The penny size of a nail indicates the length of the nail. The penny size d is given by the literal equation d = 4n – 2, where n is the length (in inches) of the nail. a. Solve the equation for n. b. Use the equation from part (a) to find the lengths of nails with the following penny sizes: 3, 6, and 10. Answer: a) The given literal equation is: d = 4n – 2 Where, n is the length ( in inches ) of the nail So, 4n = d + 2 n = ( d + 2 ) / 4 b) It is given that, The penny sizes ( d ) are: 3, 6, and 10 From part ( a ), The literal equation is: n = ( d + 2  ) / 4 Put, d= 3, 6 and 10 So, n = ( 3 + 2 ) /4 = 5 / 4 inches n = ( 6 + 2 ) / 4 = 2 inches n = ( 10 + 2 ) / 4 = 3 inches

ERROR ANALYSIS In Exercises 25 and 26, describe and correct the error in solving the equation for x.

Answer: The given literal equation is: 10 = ax – 3b So, ax = 10 + 3b x = ( 10 + 3b ) / a

In Exercises 27–30, solve the formula for the indicated variable.

Question 27. Profit: P = R – C; Solve for C.

Question 28. Surface area of a cylinder: S = 2πr 2 + 2πrh; Solve for h. Answer: The given Surface area of a cylinder is: S = 2πr² + 2πrh So, S = 2πr ( r + h ) S / 2πr = r + h h = S / 2πr – r Hence, from the above, We can conclude that the value of h is: S / ( 2π

Question 29. Area of a trapezoid: A = \(\frac{1}{2}\)h(b 1 + b 2 ); Solve for b 2 .

b. Find the mass of the pyrite sample. Answer: The mass of the pyrite sample is: 6.012 gm

Question 35. PROBLEM-SOLVING You deposit $2000 in an account that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest?

Answer: The time taken for flight is: 2.5 hours The time taken for return is: 2.3 hours

Explanation: It is given that a flight averages 460 miles per hour. The return flight averages 500 miles per hour due to a tailwind. The total flying time is 4.8 hours. We know that, Speed = Distance / Time Time = Distance / Speed It is also given that the total flying time is 4.8 hours Let the distance be D So, \(\frac{D}{460}\) + \(\frac{D}{500}\) = 4.8 \(\frac{460 + 500}{230,000}\) = 4.8 / D \(\frac{960}{230,000}\) = 4.8 / D D = 4.8 × \(\frac{230,000}{960}\) D = 1,150 miles Hence, The time taken for flight = 1,150 ÷ 460 = 2.5 hpurs The time taken for return = 1,150 ÷ 500 = 2.3 hours

b. Solve the equation for g. Answer: From the given figure, d = 20g d = 55t From part (a), t / g = 4 / 11 11t = 4g g = 11t / 4 Hence, from the above, We can conclude that the value of g is: 11t / 4

c. You travel for 6 hours. How many gallons of gasoline does the car use? How far do you travel? Explain. Answer: From part (b), g = 11t / 4 Where, g is the number of gallons of gasoline It is given that you travel for 6 hours So, t = 6 hours Now, g = ( 11 × 6 ) / 4 g = 66/4 gallons Hence, from the above, We can conclude that the number of gallons of gasoline is: 66 / 4 gallons

b. Your teacher asks you to rewrite the formula by solving for one of the side lengths, b or ℓ. Which side length would you choose? Explain your reasoning. Answer: From part (a), The surface area of the rectangular prism ( S ) = 2 ( lb + bh + lh ) S / 2 = lb + bh + lh S / 2 = b ( l + h ) + bh S / 2 = b ( l + b + h ) b = S / 2 ( l + b + h ) Hence, from the above, We can conclude that the value of b is: S / 2 ( l + b + h )

MATHEMATICAL CONNECTIONS In Exercises 43 and 44, write a formula for the area of the regular polygon. Solve the formula for the height h.

Answer: The value of h is: A / 3b

REASONING In Exercises 45 and 46, solve the literal equation for a.

Explanation: The given literal equation is: y = x [ \(\frac{ab}{a – b}\) \(\frac{ab}{a – b}\) = y / x x ( ab ) = y ( a – b ) abx = ay – by by = ay – abx by = a ( y – bx ) a = \(\frac{by}{y – bx}\) Hence, from the above, We can conclude that the value of a is: \(\frac{by}{y – bx}\)

Evaluate the expression.

Question 47. 15 – 5 + 5 2

Question 48. 18 • 2 – 4 2 ÷ 8 Answer: The given expression is: 18 ⋅ 2 – 4² ÷ 8 We have to remember that, When there is an expression to solve with multiple mathematical symbols, we have to follow the BODMAS rule BODMAS indicates the hierarchy we have to follow when we will solve mathematical symbols In BODMAS, B – Brackets O – Of D – Division M – Multiplication A – Addition S – Subtraction So, 18 ⋅ 2 – 4² ÷ 8 = 18 ⋅ 2 – ( 4 × 4 ) ÷ 8 = 18 ⋅ 2 – 2 = 36 – 2 = 34

Question 49. 3 3 + 12 ÷ 3 • 5

Question 50. 2 5 (5 – 6) + 9 ÷ 3 Answer: The given expression is: 2 5 (5 – 6) + 9 ÷ 3 We have to remember that, When there is an expression to solve with multiple mathematical symbols, we have to follow the BODMAS rule BODMAS indicates the hierarchy we have to follow when we will solve mathematical symbols In BODMAS, B – Brackets O – Of D – Division M – Multiplication A – Addition S – Subtraction So, 2 5 (5 – 6) + 9 ÷ 3 = ( 2 × 2 × 2 × 2 × 2 ) ( 5 – 6 ) + ( 9 ÷ 3 ) = ( 2 × 2 × 2 × 2 × 2 ) ( 5 – 6 ) + 3 = ( 2 × 2 × 2 × 2 × 2 ) ( -1 ) + 3 = -( 2 × 2 × 2 × 2 × 2 )  + 3 = -32 + 3 = -29

Solve the equation. Graph the solutions, if possible. (Section 1.4)

Question 51. | x – 3 | + 4 = 9

Question 52. | 3y – 12 | – 7 = 2 Answer: The values of y are: 7 and 1

Explanation: The given absolute value equation is: | 3y – 12 | – 7 = 2 | 3y – 12 | = 2 + 7 | 3y – 12 | = 9 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 3y – 12 = 9                      3y – 12 = -9 3y = 9 + 12                     3y = -9 + 12 3y = 21                            3y = 3 y = 21 / 3                         y = 3 / 3 y = 7                                y = 1 Hence, from the above, We can conclude that the values of y are: 7 and 1

Question 53. 2 | 2r + 4 | = -16

Question 54. -4 | s + 9 | = -24 Answer: The value of s is: -3 and -15

Explanation: The given absolute value equation is: -4 | s + 9 | = -24 | s + 9 | = -24 ÷ ( -4 ) | s + 9 | = 6 [ Since – ÷ – = + ] We know that, | x | = x for x > 0 | x | = -x for x < 0 So, s + 9 = 6                                       s + 9 = -6 s = 6 – 9                                         s = -6 – 9 s = -3                                             s = -15 Hence, from the above, We can conclude that the values of s are: -3 and -15

Solving Linear Equations Performance Task: Magic of Mathematics

1.4–1.5 What Did You Learn?

Question 1. How did you decide whether your friend’s argument in Exercise 46 on page 33 made sense? Answer: On page 33 in Exercise 46, The given absolute equation is: | 3x + 8 | – 9 = -5 | 3x + 8 | = -5 + 9 | 3x + 8 | = 4 So, from the absolute equation, We can say that the given absolute value equation has a solution But, according to your friend, The argument is that the absolute value equation has no solution

Question 2. How did you use the structure of the equation in Exercise 59 on page 34 to rewrite the equation? Answer: The given absolute value equation in Exercise 59 on page 34 is: 8 | x + 2 | – 6 = 5 | x + 2 | + 3 The above equation can be re-written as: 8 | x + 2  | – 5  | x + 2 | = 3 + 6 3 | x + 2  | = 9 Hence, from the above, We can conclude that the re-written form of the given absolute value equation is: 3 | x + 2 | = 9

Question 3. What entry points did you use to answer Exercises 43 and 44 on page 42? Answer: In Exercises 43 and 44 on page 42, We used the triangles as an entry point In Exercise 43, The given figure is a pentagon Using the above entry point, We divided the pentagon into 5 triangles In Exercise 44, The given figure is a Hexagon Using the above entry point, We divided the hexagon into 6 triangles.

Performance Task

Magic of Mathematics

1.1 Solving Simple Equations (pp. 3–10)

Question 1. z + 3 = -6 Answer: The value of z is: -9

Explanation: The given equation is: z + 3 = -6 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, z = -6 – ( +3 ) z = -6 – 3 z = -9 Hence, from the above, We can conclude that the value of z is: -9

Question 2. 2.6 = -0.2t Answer: The value of t is: -13

Explanation: The given equation is: 2.6 = -0.2t \(\frac{26}{10}\) = –\(\frac{2}{10}\)t t = \(\frac{26}{10}\) ÷ ( –\(\frac{2}{10}\) ) t = – \(\frac{26}{10}\) × \(\frac{10}{2}\) t = -13 Hence, from the above, We can conclude that the value of t is: -13

Question 3. – \(\frac{n}{5}\) = -2 Answer: The value of n is: 10

Explanation: The given equation is: –\(\frac{n}{5}\) = -2 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -n = -2 × 5 -n = -10 n = 10 Hence, from the above, We can conclude that the value of n is: 10

1.2 Solving Multi-Step Equations (pp. 11–18)

Solve the equation. Check your Solution.

Question 4. 3y + 11 = -16 Answer: The value of y is: -9

Explanation: The given equation is: 3y + 11 = -16 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 3y = -16 – 11 3y = -27 y = -27 ÷ 3 y = -9 Hence, from the above, We can conclude that the value of y is: -9

Question 5. 6 = 1 – b Answer: The value of b is: -5

Explanation: The given equation is: 6 = 1 – b When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, b = 1 – 6 b = -5 Hence, from the above, We can conclude that the value of b is: -5

Question 6. n + 5n + 7 = 43 Answer: The value of n is: 6

Explanation: The given equation is: n + 5n + 7 = 43 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 6n + 7 = 43 6n = 43 – 7 6n = 36 n = 36 ÷ 6 n = 6 Hence, from the above, We can conclude that the value of n is: 6

Question 7. -4(2z + 6) – 12 = 4 Answer: The value of z is: -5

Explanation: The given equation is: -4 ( 2z + 6 ) – 12 = 4 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, -4 ( 2z + 6 ) = 4 + 12 -4 ( 2z + 6 ) = 16 -4 ( 2z ) – 4 ( 6 ) = 16 -8z – 24 = 16 -8z = 16 + 24 -8z = 40 z = 40 ÷ ( -8 ) z = -5 Hence, from the above, We can conclude that the value of z is: -5

Question 8. \(\frac{3}{2}\)(x – 2) – 5 = 19 Answer: The value of x is: 18

Explanation: The given equation is: \(\frac{3}{2}\) ( x – 2 ) – 5 = 19 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, \(\frac{3}{2}\) ( x – 2 ) = 19 + 5 \(\frac{3}{2}\) ( x – 2 ) = 24 x – 2 = 24 × \(\frac{2}{3}\) x – 2 = \(\frac{24}{1}\) × \(\frac{2}{3}\) x – 2 = 16 x = 16 + 2 x = 18 Hence, from the above, We can conclude that the value of x is: 18

Question 9. 6 = \(\frac{1}{5}\)w + \(\frac{7}{5}\)w – 4 Answer: The value of w is: \(\frac{25}{4}\)

Explanation: The given equation is: 6 = \(\frac{1}{5}\)w + \(\frac{7}{5}\)w – 4 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 6 + 4 = \(\frac{1}{5}\)w + \(\frac{7}{5}\)w 10 = w [ \(\frac{1 + 7}{5}\) ] 10 = \(\frac{8}{5}\)w w = 10 × \(\frac{5}{8}\) w = \(\frac{10}{1}\) × \(\frac{5}{8}\) w = \(\frac{25}{4}\) Hence, from the above, We can conclude that the value of w is: \(\frac{25}{4}\)

Find the value of x. Then find the angle measures of the polygon.

Answer: The angle measures of the given polygon are: 110°, 50°, 20°

Answer: The angle measures of the given polygon are: 126°, 126°, 96°, 96°, 96°

1.3 Solving Equations with Variables on Both Sides (pp. 19–24)

Question 12. 3n – 3 = 4n + 1 Answer: The value of n is: -4

Explanation: The given equation is: 3n – 3 = 4n + 1 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4n – 3n = -1 – 3 n = -4 Hence, from the above, We can conclude that the value of n is: -4

Question 13. 5(1 + x) = 5x + 5 Answer: The given equation has no solution

Explanation: The given equation is: 5 ( 1 + x ) = 5x + 5 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 5 ( 1 ) + 5 ( x ) = 5x + 5 5 + 5x = 5x + 5 5 = 5x – 5x + 5 5 = 5 Hence, from the above, We can conclude that the given equation has no solution

Question 14. 3(n + 4) = \(\frac{1}{2}\)(6n + 4) Answer: The given equation has no solution

Explanation: The given equation is: 3 ( n + 4 ) = \(\frac{1}{2}\) ( 6n + 4 ) When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 3 ( n ) + 3 ( 4 ) = \(\frac{1}{2}\) ( 6n + 4 ) 3n + 12 = \(\frac{1}{2}\) ( 6n + 4 ) 2 ( 3n + 12 ) = 6n + 4 2 ( 3n ) + 2 ( 12 ) = 6n + 4 6n + 24 = 6n + 4 24 = 6n – 6n + 4 24 = 4 Hence, from the above, We can conclude that the given equation has no solution

1.4 Solving Absolute Value Equations (pp. 27–34)

Check the apparent solutions to see if either is extraneous. The solution is x = 3. Reject x = -1 because it is extraneous.

Question 15. | y + 3 | = 17 Answer: The value of y is: 14 or -20

Explaantion: The given absolute value equation is: | y + 3 | = 17 We know that, | x | = x for x  0 | x | = -x for x < 0 So, y + 3 = 17                               y + 3 = -17 y = 17 – 3                                y = -17 – 3 y = 14                                      y = -20 Hence, from the above, We can conclude that the value of y is: 14 or -20

Question 16. -2 | 5w – 7 | + 9 = -7 Answer: The value of w is: 3 or –\(\frac{1}{5}\)

Explanation: The given absolute value equation is: -2 | 5w – 7 | + 9 = -7 -2 | 5w – 7 | = -7 – 9 -2 | 5w – 7 | = -16 | 5w – 7 | = -16 / ( -2 ) | 5w – 7 | = 8 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 5w – 7 = 8                                     5w – 7 = -8 5w = 8 + 7                                    5w = -8 + 7 5w = 15                                         5w = -1 w = 15 ÷ 5                                     w = –\(\frac{1}{5}\) w = 3                                              w =-\(\frac{1}{5}\) Hence, from the above, We can conclude that the value of w is: 3 or –\(\frac{1}{5}\)

Question 17. | x – 2 | = | 4 + x | Answer: The given absolute equation has no solution

Explanation: The given absolute value equation is: | x – 2 | = | 4 + x | We know that, | x | = x for x > 0 | x | = -x for x < 0 So, x – 2 = 4 + x                                                   – ( x – 2 ) = – ( 4 + x ) -2 = 4                                                                2 = -4 Hence, from the above, We can conclude that the given absolute equation has no solution

Question 18. The minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum sustained wind speed is 95 miles per hour. Write an absolute value equation that represents the minimum and maximum speeds. Answer: The absolute value equation that represents the minimum and maximum speeds is: | x – 84.5 | = 9.5

Explanation: It is given that the minimum sustained wind speed of a Category 1 hurricane is 74 miles per hour. The maximum sustained wind speed is 95 miles per hour. So, The average wind speed sustained = ( 74 + 95 ) /2 = 169 / 2 = 84.5 miles per hour Now, The minimum wind speed from the average speed point = 84.5 – 74 = 9.5 miles per hour So, The absolute value equation that represents the minimum and maximum wind speed is: | x – 84.5 | = 9.5

Question 19. 2x – 4y = 20 Answer: The value of y is: ( x / 2 ) – 5

Explanation: The given literal equation is: 2x – 4y = 20 When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4y = 2x – 20 y = ( 2x – 20 ) / 4 y = ( 2x / 4 ) – ( 20 / 4 ) y = ( x / 2 ) – 5 Hence, from the above, We can conclude that the value of y is: ( x / 2 ) – 5

Question 20. 8x – 3 = 5 + 4y Answer: The value of y is: 2x – 2

Explanation: The given literal equation is: 8x – 3 = 5 + 4y When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, 4y = 8x – 3 – 5 4y = 8x – 8 y = ( 8x – 8 ) / 4 y = ( 8x / 4 ) – ( 8 – 4 ) y = 2x – 2 Hence, from the above, We can conclude that the value of y is: 2x – 2

Question 21. a a = 9y + 3yx Answer: The value of y is: a² / ( 3x + 9 )

Explanation: The given literal equation is: a² = 9y + 3yx When we convert any sign from LHS, then the sign will be converted into the opposite sign in RHS, So, + in LHS is converted into – in RHS and vice-versa × in LHS is converted into ÷ in RHS and vice-versa So, a² = y ( 3x + 9 ) y = a² / ( 3x + 9 ) Hence, from the above, We can conclude that the value of y is: a² / ( 3x + 9 )

Answer: a) The value of h is: \(\frac{3V}{B}\)

Explanation: The given formula is: V = \(\frac{1}{3}\)Bh Where, B is the area of the base h is the height Now, 3V = Bh h = \(\frac{3V}{B}\) Hence, from the above, We can conclude that the value of h is: \(\frac{3V}{B}\)

b) The value of h is: 18 cm

Explanation: From the given figure, Area of the base ( B ) = 36 cm² Volume of the base ( V ) = 216 cm³ From part (a), h = \(\frac{3V}{B}\) h = \(\frac{3 × 216}{36}\) h = \(\frac{3 × 216}{36 × 1}\) h = 18 cm Hence, from the above, We can conclud ethat the value of h is: 18 cm

Question 23. The formula F = \(\frac{9}{5}\)(K – 273.15) + 32 converts a temperature from kelvin K to degrees Fahrenheit F. a. Solve the formula for K. b. Convert 180°F to kelvin K. Round your answer to the nearest hundredth. Answer: a) The formula for K is: K = \(\frac{5}{9}\) ( F – 32 ) + 273.15

Explanation: The given formula for F is: F = \(\frac{9}{5}\) ( K – 273.15 ) + 32 Now, F – 32 = \(\frac{9}{5}\) ( K – 273.15 ) \(\frac{5}{9}\) ( F – 32 ) = K – 273.15 K = \(\frac{5}{9}\) ( F – 32 ) + 273.15 Hence, from the above, We can conclude that the value of K is: \(\frac{5}{9}\) ( F – 32 ) + 273.15

Question 1. x – 7 = 15 Answer: The value of x is: 22

Explanation: The given equation is: x – 7 = 15 Now, x = 15 + 7 x = 22 Hence, from the above, We can conclude that the value of x is: 22

Question 2. \(\frac{2}{3}\)x = 5 Answer: The value of x is: \(\frac{15}{2}\)

Explanation: The given equation is: \(\frac{2}{3}\) x = 5 Now, x = 5 × \(\frac{3}{2}\) x = \(\frac{5}{1}\) × \(\frac{3}{2}\) x = \(\frac{15}{2}\) Hence, from the above, We can conclude that the value of x is: \(\frac{15}{2}\)

Question 3 11x + 1 = -1 + x Answer: The value of x is: –\(\frac{1}{5}\)

Explanation: The given equation is: 11x + 1 = -1 + x Now, 11x – x = -1 – 1 10x = -2 x = –\(\frac{2}{10}\) x = –\(\frac{1}{5}\) Hence, from the above, We can conclude that the value of x is: –\(\frac{1}{2}\)

Question 4. 2 | x – 3 | – 5 = 7 Answer: The value of x is: 9 or -3

Explanation: The given absolute value equation is: 2 | x – 3 | – 5 = 7 2 | x – 3 | = 7 + 5 2 | x – 3 | = 12 | x – 3 | = \(\frac{12}{2}\) | x – 3 | = 6 We know that, | x | = x for x > 0 | x | = -x for x < 0 So, x – 3 = 6                             x – 3 = -6 x = 6 + 3                            x = -6 + 3 x = 9                                   x = -3 Hence, from the above, We can conclude that the value of x is: 9 or -3

Question 5. | 2x – 19 | = 4x + 1 Answer: The value of x is: -10 or 3

Explanation: The given absolute value equation is: | 2x – 19 | = 4x + 1 We know that, | x | = x for x > 0 | x | = -x for x < 0 4x + 1 = 2x – 19                         4x + 1 = – ( 2x – 19 ) 4x – 2x = -19 – 1                         4x + 2x = 19 – 1 2x = -20                                      6x = 18 x = \(\frac{-20}{2}\)      x = \(\frac{18}{6}\) x = -10                                         x = 3 Hence, from the above, We can conclude that the value of x is: -10 or 3

Question 6. -2 + 5x – 7 = 3x – 9 + 2x Answer: The given absolute equation has no solution

Explanation: The given equation is: -2 + 5x – 7 = 3x – 9 + 2x 5x – 9 = 5x – 9 Hence, from the above, We can conclude that the given absolute value equation has no solution

Question 7. 3(x + 4) – 1 = -7 Answer: The value of x is: -6

Explanation: The given equation is: 3 ( x + 4 ) – 1 = -7 So, 3 ( x ) + 3 ( 4 ) = -7 + 1 3x + 12 = -6 3x = -6 – 12 3x = -18 x = –\(\frac{18}{3}\) x = -6 Hence, from the above, We can conclude that the value of x is: -6

Question 8. | 20 + 2x | = | 4x + 4 | Answer: The value of x is: 8

Explanation: The given absolute value equation is: | 20 + 2x | = | 4x + 4 | We know that, | x | = x for x > 0 | x | = -x for x < 0 So, 20 + 2x = 4x + 4 4x – 2x = 20 – 4 2x = 16 x = \(\frac{16}{2}\) x = 8 Hence, from the above, We can conclude that the value of x is: 8

Question 9. \(\frac{1}{3}\)(6x + 12) – 2(x – 7) = 19 Answer: The given equation has no solution

Explanation: The given equation is: \(\frac{1}{3}\) ( 6x + 12 ) – 2 ( x – 7 ) = 19 Now, \(\frac{1}{3}\) ( 6x – 12 ) = 19 + 2 ( x – 7 ) \(\frac{1}{3}\) ( 6x – 12 ) = 19 + 2x – 14 \(\frac{1}{3}\) ( 6x – 12 ) = 2x + 5 1 ( 6x – 12 ) = 3 ( 2x + 5 ) 6x – 12 = 6x + 15 6x – 6x = 15 + 12 15 = -12 Hence, from the above, We can conclude that the given equation has no solution

Describe the values of c for which the equation has no solution. Explain your reasoning.

Question 10. 3x – 5 = 3x – c Answer: The value of c is: 5

Explanation: The given equation is: 3x – 5 = 3x – c It is given that the equation has no solution So, 3x – 3x – 5 =-c -c = -5 c = 5 Hence, from the above, We can conclude that the value of c is: 5

Question 11. | x – 7 | = c Answer: The value of c is: -7

Explanation: The given absolute value equation is: | x – 7 | = c It is given that the equation has no solution i.e., x = 0 So, 0 – 7 = c c = -7 Hence, from the above, We can conclude that the value of c is: -7

Question 12. A safety regulation states that the minimum height of a handrail is 30 inches. The maximum height is 38 inches. Write an absolute value equation that represents the minimum and maximum heights. Answer: The absolute value expression that represents the minimum and maximum heights is: | x – 64 | = 34

Explanation: It is given that a safety regulation states that the minimum height of a handrail is 30 inches. The maximum height is 38 inches. So, The average height of a handrail = ( 30 + 38 ) / 2 = 68 / 2 = 34 inches Now, The minimum height from the average height of a handrail = 34 + 30 = 64 inches Hence, The absolute value equation that represents the minimum and maximum height of a handrail is: | x – 64 | = 34

Question 13. The perimeter P (in yards) of a soccer field is represented by the formula P = 2ℓ + 2w, where ℓ is the length (in yards) and w is the width (in yards). a. Solve the formula for w. Answer: The given formula is: P = 2l + 2w Where, P is perimeter ( in yards ) l is the length ( in yards ) w is the width ( in yards ) So, 2w = P – 2l w = ( P – 2l ) / 2 Hence, from the above, We can conclude that the formula for w is: w = ( P – 2l ) / 2

Question 15. Consider the equation | 4x + 20 | = 6x. Without calculating, how do you know that x = -2 is an extraneous solution? Answer: We know that, The absolute value equations only accept the values greater than or equal to 0 Hence, For the given absolute value equation, | 4x + 20 | = 6x x = -2 is an extraneous solution

Question 16. Your friend was solving the equation shown and was confused by the result “-8 = -8.” Explain what this result means. 4(y – 2) – 2y = 6y – 8 – 4y 4y – 8 – 2y = 6y – 8 – 4y 2y – 8 = 2y – 8 -8 = -8 Answer: The result ” -8 = -8 ” means that the solved equation has no solution

Question 1. A mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are divided evenly between intermediate and expert trials. How many of each kind of trail are there? A. 12 beginner, 18 intermediate, 18 expert B. 18 beginner, 15 intermediate, 15 expert C. 18 beginner, 12 intermediate, 18 expert D. 30 beginner, 9 intermediate, 9 expert

Answer: The correct option is: B The number of beginner trials is: 18 The number of intermediate trials is: 15 The number of expert trials is: 15

Explanation: It is given that a mountain biking park has 48 trails, 37.5% of which are beginner trails. The rest are divided evenly between intermediate and expert trials. So, The number of beginner trials is 3.5 % of the total number of trials It is given that the total number of trials is: 48 We know that, The value of 37.5 % is: \(\frac{3}{8}\) [ 37.5 % = 50 % – 12.5 % ] So, The number of beginner trials = \(\frac{3}{8}\) × 48 = \(\frac{3}{8}\) × \(\frac{48}{1}\) = \(\frac{3 × 48}{8 × 1}\) = 18 So, The number  of intermediate and expert trials = ( The total number of trials ) – ( The number of beginner trials ) = 48 – 18 = 30 trials It is also given that the intermediate trials and expert trails are divided evenly So, 30 ÷ 2 = 15 trials each Hence, from the above, We can conclude that The number of beginner trials is: 18 The number of intermediate trials is: 15 The number of expert trials is: 15

Answer: The given equations are: a) cx – a + b = 2b b) 0 = cx – a + b c) 2cx – 2a = b / 2 d) x – a = b / 2 e) x = ( a + b ) / c f) b + a = cx Now, We have to find the equations from above that is equivalent to the given equation cx – a = b Now, a) The given equation is: cx – a + b = 2b So, cx – a = b – b cx – a = b b) The given equation is: 0 = cx – a + b So, cx – a = -b c) The given equation is: 2cx – 2a = b / 2 So, cx – a = b / 4 d ) The given equation is: cx – a = b / 2 So, 2 ( cx – a ) = b e) The given equation is: x = ( a + b ) / c So, cx = a + b cx – a = b f) The given equation is: b + a = cx So, cx – a = b Hence, from the above, We can conclude that the equations that are equivalent to cx – a = b is: a, e, f

Question 3. Let N represent the number of solutions of the equation 3(x – a) = 3x – 6. Complete each statement with the symbol <, >, or =. a. When a = 3, N ____ 1. b. When a = -3, N ____ 1. c. When a = 2, N ____ 1. d. When a = -2, N ____ 1. e. When a = x, N ____ 1. f. When a = -x, N ____ 1.

Answer: The given equation is: 3 ( x – a ) = 3x – 6 So, 3x – 3a = 3x – 6 Now, a) When a = 3, 3x – 3 ( 3 ) = 3x – 6 3x – 9 = 3x – 6 9 = 6 Hence, When a = 3 there is no solution Hence, N < 1 b) When a = -3 3x + 3 ( 3 ) =3 x – 6 9 = -6 Hence, When a = -3, there is no solution Hence, N < 1 c) When a = 2 3x – 3 ( 2 ) = 3x – 6 3x – 6 =3x – 6 = 6 Hence, When a= 2, there is no solution Hence, N < 1 d) When a = -2 3x + 3 ( 2 ) = 3x – 6 3x + 6 = 3x – 6 6 = -6 Hence, When a = -2, thereis no solution Hence, N < 1 e) When a = x 3x – 3 ( x ) = 3x – 6 3x = 6 x = 6 / 3 x = 2 Hence, When a  x, theer is 1 solution Hence, N = 1 f) When a = -x 3x + 3 ( x ) = 3x – 6 6x – 3x = -6 3x = -6 x = -6 / 3 x = -2 Hence, When a = -x, there is 1 solution Hence, N = 1

Explanation: It is given that you spend $132 on 5 cans of paint It is also given that the white paint costs $24 per can and the blue paint costs $28 per can Now, Let The number of white cans is: x The number of blue cans be: 5 – x So, The total cost of paint = ( The number of white cans ) × ( The cost of white paint per can ) + ( The number of blue cans ) × ( The cost of blue paint per can ) 132 = 24x + 28 ( 5 – x ) 24x + 28 ( 5 ) – 28x = 132 140 – 4x = 132 4x = 140 – 132 4x = 8 x = 8 ÷ 4 x = 2 Hence, from the above, We can conclude that The number of cans of white paint is: 2 The number of cans of blue paint is: 3

b. How much would you have saved by switching the colors of the dining room and living room? Explain. Answer: The money you have saved by switching the colors of the dining room and living room is: $0

Explanation: From part (a), The number of white cans is: 2 The number of blue cans is: 3 It is given that white color is used in the dining room and the blue color is used in the living room So, The cost of white paint after interchanging the color in the living room = 24 × 2 = $48 The cost of blue paint after interchanging the color in the dining room = 28 × 3 = $84 So, The total cost of paint after interchanging the colors = 48 + 84 = $132 Hence, The amount of money saved = ( The money you paid for the paint before interchanging ) – ( The money you paid for the paint after interchanging ) = 132 – 132 =$0

Answer: The given equations are: a ) 6x + 6 = -14 b ) 8x + 6 = -2x – 14 c ) 5x + 3 = -7 d ) 7x + 3 = 2x – 13 Now, To find the equivalent equations, find the value of x So, a) The given equation is: 6x + 6 = -14 6x = -14 – 6 6x = -20 x = -20 / 6 x = -10 / 3 b) The given equation is: 8x + 6 = -2x – 14 8x + 2x = -14 – 6 10x = -20 x = -20 / 10 x = -2 c) The given equation is: 5x + 3 = -7 5x = -7 -3 5x = -10 x= -10 / 5 x = -2 d) The given equation is: 7x + 3 = 2x – 13 7x – 2x = -13 – 3 5x = -16 x = -16 / 5 Hence, from the above, We can conclude that the equations c) and d) are equivalent

Explanation: We know that, The perimeter is the sum of all the sides of the given figure It is given that the perimeter of the triangle is: 13 inches So, The perimeter of the triangle = ( x – 5 ) + ( x / 2 ) + 6 13 = x + 1 + ( x / 2 ) 13 = ( 2x / 2 ) + ( x / 2 ) + 1 3x / 2 = 13 – 1 3x / 2 = 12 3x = 12 × 2 3x = 24 x = 24 / 3 x = 8 So, The lengths of all sides are: ( 8 – 5 ), 6, ( 8 / 2 ) = 3 inches, 6 inches, 4 inches Hence, from the above, We can conclude that the length of the shortest side is: 3 inches

Question 7. You pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and pays $36 per month for satellite TV. Your friend claims that the expenses for a year of satellite TV are less than the expenses for a year of cable. a. Write and solve an equation to determine when you and your friend will have paid the same amount for TV services. Answer: It is given that you pay $45 per month for cable TV. Your friend buys a satellite TV receiver for $99 and pays $36 per month for satellite TV So, Let the number of months be x Now, The time they paid the same amount for TV services is: 45x = 99 + 36x 45x – 36x = 99 9x= 99 x = 99 / 9 x = 11 Hence, from the above, We can conclude that after 11 months, you and your friend will pay the same amount for TV services

b. Is your friend correct? Explain. Answer: Your friend is correct

Explanation: We know that, 1 year = 12 months So, The expenses paid by you for TV services = 45x = 45 × 12 = $540 The expenses paid by your friend for TV services = 99 + 36x = 99 + 36 ( 12 ) = 99 + 432 = $531 By comparing the expenses of you and your friend, We can conclude that your friend is correct

Answer: We know that, Average speed = ( Distance ) ÷ ( Time ) It is given that a car travels 1000 feet in 12.5 seconds So, Average speed = 1000 / 12.5 = 80\(\frac{feet}{second}\) Now, The given options are: A) 80\(\frac{second}{feet}\) B) 80\(\frac{feet}{second}\) C) \(\frac{80 feet}{second}\) D) \(\frac{second}{ 80 feet}\) Hence, from the above, We can conclude that option B) represents the average speed

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Common Core Algebra I

The full experience and value of eMATHinstruction courses are achieved when units and lessons are followed in order.  Students learn skills in earlier units that they will then build upon later in the course.  Lessons can be used in isolation but are most effective when used in conjunction with the other lessons in this course. All Lesson/Homework files, Spanish translations of those files, and videos are available for free.  Other resources, such as answer keys and more, are accessible with a paid  membership .

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Standards Alignment – Powered by EdGate

• Unit 1 - The Building Blocks of Algebra
• Table of Contents for Common Core Algebra I
• Unit 2 - Linear Expressions, Equations, and Inequalities
• Unit 3 - Functions
• Unit 4 - Linear Functions and Arithmetic Sequences
• Unit 5 - Systems of Linear Equations and Inequalities
• Unit 6 - Exponents, Exponents, Exponents and More Exponents
• Unit 7 - Polynomials
• Unit 8 - Quadratic Functions and Their Algebra
• Unit 9 - Roots and Irrational Numbers
• Unit 10 - Statistics
• Unit 11 - A Final Look at Functions and Modeling

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