Name ——————————————————————— LESSON 3.4 Date ———————————— Practice A For use with pages 154 –160 1. 10x 2 7 5 4x 1 5 2. 3x 1 6 5 22x 1 11 A. 6x 2 7 5 5 A. 5x 1 6 5 11 3. 6(3x 2 4) 5 12 A. 18x 2 24 5 12 B. 6x 5 12 B. 5x 5 5 B. 18x 5 36 C. x52 C. x51 C. x52 4. 6(x 1 3) 5 5x 1 8 5. 4(x 2 2) 5 7x 1 1 A. 6x 1 18 5 5x 1 8 B. C. 6. 2x 1 2 5 4(x 2 5) A. 4x 2 8 5 7x 1 1 x 1 18 5 8 x 5 210 LESSON 3.4 Describe each step used in solving the equation. A. 2x 1 2 5 4x 2 20 B. 28 5 3x 1 1 B. 2 5 2x 2 20 C. 29 5 3x C. 22 5 2x D. 23 5 x D. 11 5 x Solve the equation and describe each step you use. 7. 6p 2 3 5 4p 2 1 8. 10a 2 2 5 7a 1 4 9. 5(m 1 2) 5 20 Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. Solve the equation, if possible. 10. 9x 2 2 5 8x 1 7 11. 5n 2 3 5 3n 1 1 12. 4z 2 5 5 8z 1 3 13. 2a 1 4 5 a 1 6 14. w 1 8 5 w 2 3 15. 2( y 2 3) 5 y 1 4 16. 3(m 1 2) 5 8 1 m 17. 6 1 x 5 6(x 2 5) 18. 7(b 1 3) 5 7b 2 4 19. Dimensions of a Circular Flower Garden A ﬂower garden has the shape shown. The diameter of the outer circle is three times the diameter of the inner circle. The lengths of the walkways are 8 feet long. What is the diameter of the inner circle? x 8 ft 8 ft 3x 20. Distance-Rate-Time Two cars travel the same distance. The ﬁrst car travels at a rate of 50 miles per hour and reaches its destination in t hours. The second car travels at a rate of 60 miles per hour and reaches its destination 1 hour earlier than the ﬁrst car. How long does it take for the ﬁrst car to reach its destination? Rate of car 1 p Time for car 1 5 Rate of car 2 p Time for car 2 Algebra 1 Chapter 3 Resource Book 41 Name ——————————————————————— Practice B LESSON 3.4 LESSON 3.4 Date ———————————— For use with pages 154 –160 Solve the equation and describe each step you use. 1. 5x 1 11 5 4x 1 18 2. 11p 2 4 5 6p 1 1 3. 26 5 2(w 1 5) Solve the equation, if possible. 4. 15x 2 8 5 14x 1 13 5. 9n 2 7 5 5n 1 5 6. 4z 2 15 5 4z 1 11 7. 27a 1 9 5 3a 1 49 8. 4(w 1 3) 5 w 2 15 9. 8( y 2 5) 5 6y 2 18 10. 14m 2 10 5 3(4 1 m) 1 11. 7 1 x 5 } (4x 2 2) 2 12. 8b 1 11 2 3b 5 2b 1 2 13. 10d 2 6 5 4d 2 15 2 3d 14. 16p 2 4 5 4(2p 2 3) 15. 0.25(8z 2 4) 5 z 1 8 2 2z Find the perimeter of the square. 16. 17. 18. 5x ⫺ 8 3x 7x ⫺ 15 10x 6x 1 8 2x 19. Saving and Spending Currently, you have $80 and your sister has $145. You decide 20. Botanical Gardens The membership fee for joining a gardening association is $24 per year. A local botanical garden charges members of the gardening association $3 for admission to the garden. Nonmembers of the association are charged $6. After how many visits to the garden is the total cost for members, including the membership fee, the same as the total cost for nonmembers? 21. College Enrollment Information about students’ choices of majors at a small college is shown in the table. In how many years will there be 2 times as many students majoring in engineering than in business? In how many years will there be 2 times as many students majoring in engineering than in biology? Major 42 Number of students enrolled in major Average rate of change Engineering 120 22 more students each year Business 105 4 fewer students each year Biology 98 6 more students each year Algebra 1 Chapter 3 Resource Book Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. to save $6 of your allowance each week, while your sister decides to spend her whole allowance plus $7 each week. How long will it be before you have as much money as your sister? Name ——————————————————————— LESSON 3.4 Date ———————————— Practice C For use with pages 154 –160 1. 9x 2 4 5 7x 2 16 2. 5d 2 3 1 2d 5 4d 1 9 3. 4(2m 1 5) 5 3m 2 5 4. 6(7 2 2p) 5 3(5p 1 1) 5. 11w 1 2(w 1 1) 5 5w 2 6 6. 10 1 2(2a 1 1) 5 7a 2 3 1 7. } (12n 2 2) 5 5n 2 7 2 1 8. 4y 1 16 5 } (10y 2 4) 3 3 9. 2} (8x 2 12) 5 5x 2 2 4 LESSON 3.4 Solve the equation and describe each step you use. Solve the equation, if possible. 10. 42n 2 13 5 17n 1 12 11. 4.5x 1 3.4 5 1.5x 2 2.6 12. 14d 2 43 5 6d 2 13 2 7d 13. 24(2w 2 5) 5 3w 2 13 14. 14 2 4z 5 2(17 2 z) 15. 9(4h 2 6) 5 2(213 2 2h) 3 1 2 1 16. } x 1 } 5 } x 2 } 2 3 3 2 1 17. } (6x 1 3) 5 2x 2 5 3 1 1 18. } (9 2 2x) 5 } (3x 1 4) 8 4 19. 6.8t 2 10 2 3.2t 5 3t 2 1 20. 6(1.3p 2 3) 5 2.6p 2 5 21. 0.2(15z 2 5) 5 4(4z 1 1) Find the length and the width of the rectangle described. 22. The length is 5 units more than the width. The perimeter is 9 times the width. 23. The length is 5 units less than 2 times the width. The perimeter is 22 units more than twice the width. Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. 24. High School Enrollments Central High’s enrollment decreases at an average rate of 55 students per year, while Washington High’s enrollment increases at an average rate of 70 students per year. Central High has 2176 students and Washington High has 1866 students. If enrollments continue to change at the same rate, when will the two schools have the same number of students? 25. Teeter-Totter Two children weighing 42 pounds and 54 pounds are on a teeter-totter as shown. The 54-pound child is sitting 1 foot closer to the center than the 42-pound child. To balance the teeter-totter, the 42-pound child must sit x feet from the center. Write an equation to ﬁnd how many feet the 42-pound child must be from the center of the teeter-totter so it is balanced. Solve for x. 54 lb 42 lb x⫺1 x 26. Charity Race In a 5-mile charity race, groups of runners are started 5 minutes apart. One runner in the ﬁrst group is running at a rate of 0.06 mile per minute. One runner in the second group is running at a rate of 0.1 mile per minute. a. Let t represent the time (in minutes) it takes the runner from the ﬁrst group to run the race. Write and solve an equation to ﬁnd the number of minutes after which the runner from the second group would catch up with the runner from the ﬁrst group. b. Does the runner from the second group pass the runner from the ﬁrst group before the race is over? Explain your reasoning. Algebra 1 Chapter 3 Resource Book 43 Name ——————————————————————— LESSON 3.4 Date ———————————— Challenge Practice For use with pages 154 –160 LESSON 3.4 1. For what value of a is a(x 2 5) 5 9x 2 25 2 2x 2 10 an identity? 2. For what value of b is 2x 2 bx 2 3 5 b(2x 2 3) 213x 1 12 an identity? 3. For what value of c is 2(cx 1 12) 5 3(cx 1 8) an identity? 4. For what value of d is d(dx 1 1) 5 24dx 2 2d 2 12 an identity? 5. Find the area of a rectangle whose perimeter is 34 inches and whose width is two more than twice the length. 6. Find the area of a rectangle whose length is 6 inches less than 5 times the width and whose perimeter is 8 inches more than twice the length. 7. Find the area of a rectangle whose length is one-third of the perimeter, whose width is one-half of the length, and whose perimeter is 60 inches. 8. Find the length of a rectangle which when cut in half has an area of 300 square Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. inches and whose width is one-sixth of the length. Algebra 1 Chapter 3 Resource Book 49 Name ——————————————————————— LESSONS 3.1–3.4 Date ———————————— Problem Solving Workshop: Mixed Problem Solving For use with pages 134–159 rectangle is 12 inches greater than the width. The perimeter of the rectangle is 48 inches. a. Write a verbal model for the situation. a. Write and solve an equation to ﬁnd the b. Use the verbal model to write and solve dimensions of the rectangle. b. What is the area of the rectangle? c. Suppose the length and width are doubled. What affect does this have on the perimeter and area of the rectangle? an equation. How many shirts did you buy? c. What was the average cost per shirt? 2. Multi-Step Problem Your kitchen ﬂoor is 30 feet long and 18 feet wide. You want to cover the kitchen ﬂoor with square tiles that have a side length of 3 feet. a. Find the area of one tile. b. Write an equation for the number of tiles that will cover the kitchen ﬂoor. c. How many tiles are needed to cover the kitchen ﬂoor? 3. Gridded Answer Each side of the Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. 6. Extended Response The length of a for $12 per shirt. Every shirt after the second shirt is $4 off. You spent $56. triangle has the same length. The perimeter of the triangle is how many units? 2x 1 7 3x 1 4 4. Short Response A basketball team plays 20 games and averages 58 points per game. Write a verbal model for ﬁnding the total number of points for the season. Write and solve an algebraic model. The team’s goal for next year is to score 5 more points per game for 20 games. How many total points does the team want to score next year? Explain how you found your answer. 5. Gridded Answer Mike weighs 110 pounds. He weighs 23 pounds more than Brian. How many pounds does Brian weigh? LESSON 3.4 1. Multi-Step Problem You bought n shirts 7. Open-Ended Write an equation that can be solved using only addition. Write an equation that can be solved using only multiplication. Write a two-step equation that can be solved using addition and multiplication. 8. Extended Response In basketball, a player’s ﬁeld goal percentage is calculated by dividing the number of ﬁeld goals made by the number of ﬁeld goals attempted. Player Steve Nash Team Phoenix Suns Field goal percentage 0.502 Field goals attempted 857 a. Use the information in the table to ﬁnd the number of ﬁeld goals made by Steve Nash in the 2004–2005 National Basketball Association regular season. Round your answer to the nearest whole number. b. Tony Parker of the San Antonio Spurs made 109 more ﬁeld goals than Steve Nash in the 2004–2005 regular season. How many ﬁeld goals did Tony Parker make? c. In the 2004–2005 regular season, Steve Nash had a higher ﬁeld goal percentage than Tony Parker. Did Tony Parker have fewer attempts than Steve Nash? Explain your reasoning. Algebra 1 Chapter 3 Resource Book 47 Name ——————————————————————— LESSON 3.4 Date ———————————— Investigating Algebra Activity: Modeling Equations with Variables on Both Sides For use before Lesson 3.4 QUESTION EXPLORE algebra tiles How can you use algebra tiles to solve an equation with a variable on both the left and the right side of the equation? LESSON 3.4 Materials: Solve an equation with variables on both sides Solve 5x 1 4 5 3x 1 8. STEP 1 Model 5x 1 4 5 3x 1 8 using algebra tiles. STEP 2 You want to have x-tiles on only one side of the equation, so subtract three x-tiles from each side. 5 5 Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. STEP 3 To isolate the x-tiles, subtract four 1-tiles from each side. STEP 4 There are two x-tiles, so divide the x-tiles and 1-tiles into two equal groups. So, x 5 2. 5 5 DRAW CONCLUSIONS Use algebra tiles to model and solve the equation. 1. 4x 1 3 5 3x 1 7 2. 2x 1 8 5 11 1 x 3. 5x 1 9 5 8x 1 6 4. 7x 1 6 5 9x 1 2 5. Copy and complete the equations and explanations. 2x 1 19 5 7x 1 4 2x 1 19 2 ? 5 7x 1 4 2 ? 19 5 ? 19 2 ? 14 Original equation Subtract ? from each side. Simplify. 5 5x 1 4 2 ? Subtract ? from each side. ? 5 5x Simplify. ? 5x Divide each side by ? and simplify. Algebra 1 Chapter 3 Resource Book 39 Name ——————————————————————— LESSON LESSON 3.4 3.4 Date ———————————— Study Guide For use with pages 154 –160 GOAL Solve equations with variables on both sides. Vocabulary An equation that is true for all values of the variable is an identity. EXAMPLE 1 Solve an equation with variables on both sides Solve 13 2 6x 5 3x 2 14. Solution 13 2 6x 5 3x 2 14 Write original equation. 13 2 6x 1 6x 5 3x 2 14 1 6x Add 6x to each side. 13 5 9x 2 14 Simplify. 27 5 9x Add 14 to each side. 35x Divide each side by 9. The solution is 3. Check by substituting 3 for x in the original equation. 13 2 6x 5 3x 2 14 Write original equation. 13 2 6(3) 5 3(3) 2 14 Substitute 3 for x. 25 5 3(3) 2 14 Simplify left side. 25 5 25 ✓ Simplify right side. Solution checks. Exercises for Example 1 Solve the equation. Check your solution. 1. 9a 5 7a 2 8 EXAMPLE 2 2. 17 2 8b 5 3b 2 5 3. 25c 1 6 5 9 2 4c Solve an equation with grouping symbols 1 Solve 4x 2 7 5 } (9x 2 15). 3 Solution 1 4x 2 7 5 }3 (9x 2 15) Write original equation. 4x 2 7 5 3x 2 5 Distributive property x 2 7 5 25 x52 The solution is 2. 44 Algebra 1 Chapter 3 Resource Book Subtract 3x from each side. Add 7 to each side. Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. CHECK Name ——————————————————————— LESSON 3.4 Study Guide Date ———————————— continued For use with pages 154 –160 LESSON 3.4 Exercises for Example 2 Solve the equation. Check your solution. 4. 2m 2 7 5 3(m 1 8) 1 5. } (15n 1 5) 5 8n 2 9 5 3 6. 7p 2 3 5 } (8p 2 12) 4 EXAMPLE 3 Identify the number of solutions of an equation Solve the equation, if possible. a. 4(3x 2 2) 5 2(6x 1 1) b. 4(4x 2 5) 5 2(8x 2 10) Solution a. 4(3x 2 2) 5 2(6x 1 1) 12x 2 8 5 12x 1 2 Copyright © by McDougal Littell, a division of Houghton Mifﬂin Company. 12x 5 12x 1 10 Write original equation. Distributive property Add 8 to each side. The equation 12x 5 12x 1 10 is not true because the number 12x cannot be equal to 10 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation. 12x 2 12x 5 12x 1 10 212x 0 5 10 Subtract 12x from each side. Simplify. The statement 0 5 10 is not true, so the equation has no solution. b. 4(4x 2 5) 5 2(8x 2 10) 16x 2 20 5 16x 2 20 Write original equation. Distributive property Notice that the statement 16x 2 20 5 16x 2 20 is true for all values of x. So, the equation is an identity. Exercises for Example 3 Solve the equation, if possible. 7. 11x 1 7 5 10x 2 8 8. 5(3x 2 2) 5 3(5x 2 1) 1 9. } (6x 1 18) 5 3(x 1 3) 2 Algebra 1 Chapter 3 Resource Book 45

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