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2-4 Solving Equations with the Variable on Each Side Solve each equation. Check your solution. 11. 2m − 13 = −8m + 27 15. SOLUTION: SOLUTION: Check: Check: 13. 6 + 3t = 8t − 14 SOLUTION: 17. 6(n + 5) = 66 SOLUTION: Check: Check: 15. SOLUTION: 19. SOLUTION: eSolutions Manual - Powered by Cognero Page 1 2-4 Solving Equations with the Variable on Each Side 22. GEOMETRY Find the value of x so the rectangles have the same area. 19. SOLUTION: SOLUTION: The following is an expression for the area of the left rectangle. Check: The following is an expression for the area of the right rectangle. 21. 6(3a + 1) − 30 = 3(2a − 4) SOLUTION: Set the expressions equal to each other and solve for x. Check: Find the areas to check the solution. 22. GEOMETRY Find the value of x so the rectangles have the same area. 23. NUMBER THEORY Four times the lesser of two consecutive even integers is 12 less than twice the greater number. Find the integers. SOLUTION: eSolutions - Powered Cognero The Manual following is anby expression rectangle. When x = 8, both figures have an area of 96 square units. for the area of the left Page 2 SOLUTION: Let x be the first even integer and x + 2 be the next consecutive even integer. x + 2 = −27 + 2 or −25 x + 4 = −27 + 4 or −23 So, the consecutive odd integers are −27, −25, and −23. When x = 8, both figures have an area of 96 square 2-4 Solving Equations with the Variable on Each Side units. 23. NUMBER THEORY Four times the lesser of two consecutive even integers is 12 less than twice the greater number. Find the integers. Solve each equation. Check your solution. 25. 2x = 2(x − 3) SOLUTION: SOLUTION: Let x be the first even integer and x + 2 be the next consecutive even integer. Since , this equation has no solutions. To check, substitute any number for x. Check: Substitute −4 into each expression to find the even numbers. x = −4 x + 2 = −4 + 2 or −2 So, the consecutive even integers are −4 and −2. 27. −5(3 − q) + 4 = 5q − 11 SOLUTION: 24. CCSS SENSE-MAKING Two times the least of three consecutive odd integers exceeds three times the greatest by 15. What are the integers? SOLUTION: Let x = the least odd integer. Then x + 2 = the next greater odd integer, and x + 4 = the greatest of the three integers. Since the expressions on each side are the same, the equation is an identity. So, all numbers will work in this equation. To check, substitute any number for q. Check: 29. Substitute −27 into each expression to find the odd numbers. SOLUTION: x = −27 x + 2 = −27 + 2 or −25 x + 4 = −27 + 4 or −23 So, the consecutive odd integers are −27, −25, and −23. eSolutions Manual - Powered by Cognero Solve each equation. Check your solution. 25. 2x = 2(x − 3) Page 3 Check: 2-4 Solving Equations with the Variable on Each Side 29. 31. SOLUTION: SOLUTION: Check: Check: 31. SOLUTION: 33. 6.78j − 5.2 = 4.33j + 2.15 SOLUTION: Check: Check: 35. 3.2k − 4.3 = 12.6k + 14.5 eSolutions Manual - Powered by Cognero 33. 6.78j − 5.2 = 4.33j + 2.15 SOLUTION: Page 4 2-4 Solving Equations with the Variable on Each Side 35. 3.2k − 4.3 = 12.6k + 14.5 SOLUTION: Check: numbers. x = −2 x + 2 = −2 + 2 or 0 So, the consecutive even integers are −2 and 0. 38. MONEY Chris has saved twice the number of quarters that Nora saved plus 6. The number of quarters Chris saved is also five times the difference of the number of quarters and 3 that Nora has saved. Write and solve an equation to find the number of quarters they each have saved. SOLUTION: Let q be the number of quarters Nora saved. 37. NUMBER THEORY Three times the lesser of two consecutive even integers is 6 less than six times the greater number. Find the integers. SOLUTION: Let x be the first consecutive even integer and x + 2 be the next consecutive even integer. Substitute 7 for q to find the number of quarters Chris saved. Substitute −2 into each expression to find the even numbers. x = −2 x + 2 = −2 + 2 or 0 So, the consecutive even integers are −2 and 0. 38. MONEY Chris has saved twice the number of quarters that Nora saved plus 6. The number of quarters Chris saved is also five times the difference of the number of quarters and 3 that Nora has saved. Write and solve an equation to find the number of quarters they each have saved. SOLUTION: Let q be the number of quarters Nora saved. So, Nora saved 7 quarters and Chris saved 20 quarters. 39. DVD A company that replicates DVDs spends $1500 per day in building overhead plus $0.80 per DVD in supplies and labor. If the DVDs sell for $1.59 per disk, how many DVDs must the company sell each day before it makes a profit? SOLUTION: Let d be the number of DVDs. So, the company must sell more than 1,899 DVDs per day to make a profit. 42. REASONING Solve 5x + 2 = ax – 1 for x. Assume that a ≠ 0. Describe each step. eSolutions Manual - Powered by Cognero SOLUTION: Page 5 So, the company must sellthe more than 1,899 DVDsSide 2-4 Solving Equations with Variable on Each per day to make a profit. The correct answer is g = 6. b. This is correct. Check the answer. 42. REASONING Solve 5x + 2 = ax – 1 for x. Assume that a ≠ 0. Describe each step. SOLUTION: c. This is incorrect. To eliminate –6z on the left side of the equals sign, 6z must be added to each side of the equation. 45. CCSS CRITIQUE Determine whether each solution is correct. If the solution is not correct, describe the error and give the correct solution. a. The correct answer is z = 1. b. c. SOLUTION: a. This is incorrect. The 2 must be distributed over both g and 5. The correct answer is g = 6. b. This is correct. Check the answer. eSolutions Manual - Powered by Cognero Page 6