Practice A. Name Date. Time for car 2. Time for car 1. Rate of car 1. Rate of car 2. Describe each step used in solving the equation.

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1 Practice A For use with pages Describe each step used in solving the equation.. 0x x x 6 522x 3. 6(3x 2 4) 5 2 A. 6x A. 5x 6 5 A. 8x B. 6x 5 2 B. 5x 5 5 B. 8x 5 36 C. x 5 2 C. x 5 C. x (x 3) 5 5x (x 2 2) 5 7x 6. 2x 2 5 4(x 2 5) A. 6x 8 5 5x 8 A. 4x x A. 2x 2 5 4x 2 20 B. x B x B x 2 20 C. x 520 C x C x D x D. 5 x Solve the equation and describe each step you use. 7. 6p p a a (m 2) x x 7. 5n n 2. 4z z a 4 5 a 6 4. w 8 5 w (y 2 3) 5 y (m 2) 5 8 m 7. 6 x 5 6(x 2 5) 8. 7(b 3) 5 7b 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? 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 hour earlier than the first car. How long does it take for the first car to reach its destination? 8 ft x 8 ft 3x Rate of car p Time for car 5 Rate of car 2 p Time for car 2 4

2 Practice B For use with pages Solve the equation and describe each step you use.. 5x 5 4x 8 2. p p (w 5) 4. 5x x n n z z 7. 27a 9 5 3a (w 3) 5 w ( y 2 5) 5 6y m (4 m). 7 x 5 } 2 (4x 2 2) 2. 8b 2 3b 5 2b d d d 4. 6p (2p 2 3) (8z 2 4) 5 z 8 2 2z Find the perimeter of the square x 8 0x 7x 5 3x 6x 8 2x 9. Saving and Spending Currently, you have $80 and your sister has $45. You decide 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? 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? 2. 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 Number of students enrolled in major Average rate of change Engineering more students each year Business 05 4 fewer students each year Biology 98 6 more students each year 42

3 Practice C For use with pages Solve the equation and describe each step you use.. 9x x d 2 3 2d 5 4d (2m 5) 5 3m (7 2 2p) 5 3(5p ) 5. w 2(w ) 5 5w (2a ) 5 7a } 2 (2n 2 2) 5 5n y 6 5 } 3 (0y 2 4) } 4 (8x 2 2) 5 5x n n x 5.5x d d d 3. 24(2w 2 5) 5 3w z 5 2(7 2 z) 5. 9(4h 2 6) 5 2(23 2 2h) 6. } 2 x 2 } 3 5 } 3 x 2 3 } 2 7. } 3 (6x 3) 5 2x } 4 (9 2 2x) 5 } 8 (3x 4) t t 5 3t (.3p 2 3) 5 2.6p (5z 2 5) 5 4(4z ) 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. 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 276 students and Washington High has 866 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 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 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. 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. x 42 lb 43

4 Challenge Practice For use with pages For what value of a is a(x 2 5) 5 9x x 2 0 an identity? 2. For what value of b is 2x 2 bx b(2x 2 3) 23x 2 an identity? 3. For what value of c is 2(cx 2) 5 3(cx 8) an identity? 4. For what value of d is d(dx ) 524dx 2 2d 2 2 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 inches and whose width is one-sixth of the length. 49

5 S 3. Problem Solving Workshop: Mixed Problem Solving For use with pages Multi-Step Problem You bought n shirts for $2 per shirt. Every shirt after the second shirt is $4 off. You spent $56. a. Write a verbal model for the situation. b. Use the verbal model to write and solve 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 8 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 triangle has the same length. The perimeter of the triangle is how many units? 2x 7 3x 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 0 pounds. He weighs 23 pounds more than Brian. How many pounds does Brian weigh? 6. Extended Response The length of a rectangle is 2 inches greater than the width. The perimeter of the rectangle is 48 inches. a. Write and solve an equation to find the 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? 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 Team Steve Nash Field goal percentage Field goals attempted 857 Phoenix Suns a. Use the information in the table to find the number of field goals made by Steve Nash in the National Basketball Association regular season. Round your answer to the nearest whole number. b. Tony Parker of the San Antonio Spurs made 09 more field goals than Steve Nash in the regular season. How many field goals did Tony Parker make? c. In the 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. 47

6 QUESTION Investigating Algebra Activity: Modeling Equations with Variables on Both Sides For use before Lesson Materials: 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? EXPLORE Solve an equation with variables on both sides Solve 5x 4 5 3x 8. STEP Model 5x 4 5 3x 8 STEP 2 You want to have x-tiles on using algebra tiles. only one side of the equation, so subtract three x-tiles from each side. 5 5 DRAW CONCLUSIONS STEP 3 To isolate the x-tiles, subtract STEP 4 There are two x-tiles, so four -tiles from each side. divide the x-tiles and -tiles into two equal groups. So, x Use algebra tiles to model and solve the equation.. 4x 3 5 3x x 8 5 x 3. 5x 9 5 8x x 6 5 9x 2 5. Copy and complete the equations and explanations. 2x 9 5 7x 4 Original equation 2x 9 2? 5 7x 4 2? Subtract? from each side. 9 5? 4 Simplify. 9 2? 5 5x 4 2? Subtract? from each side.? 5 5x Simplify.? 5 x Divide each side by? and simplify. 5 39

7 Study Guide For use with pages GOAL Solve equations with variables on both sides. Vocabulary An equation that is true for all values of the variable is an identity. EXAMPLE Solve an equation with variables on both sides Solve 3 2 6x 5 3x 2 4. EXAMPLE 2 Solution 3 2 6x 5 3x 2 4 Write original equation x 6x 5 3x 2 4 6x Add 6x to each side x 2 4 Simplify x Add 4 to each side. 3 5 x Divide each side by 9. The solution is 3. Check by substituting 3 for x in the original equation. CHECK 3 2 6x 5 3x 2 4 Write original equation (3) 5 3(3) 2 4 Substitute 3 for x (3) 2 4 Simplify left side Simplify right side. Solution checks. Exercises for Example Solve the equation. Check your solution.. 9a 5 7a b 5 3b c c Solve an equation with grouping symbols Solve 4x } (9x 2 5). 3 Solution 4x } 3 (9x 2 5) Write original equation. 4x x 2 5 Distributive property x Subtract 3x from each side. x 5 2 Add 7 to each side. The solution is 2. 44

8 Study Guide continued For use with pages Exercises for Example 2 Solve the equation. Check your solution. 4. 2m (m 8) 5. } 5 (5n 5) 5 8n p } 4 (8p 2 2) EXAMPLE 3 Identify the number of solutions of an equation a. 4(3x 2 2) 5 2(6x ) b. 4(4x 2 5) 5 2(8x 2 0) Solution a. 4(3x 2 2) 5 2(6x ) Write original equation. 2x x 2 Distributive property 2x 5 2x 0 Add 8 to each side. The equation 2x 5 2x 0 is not true because the number 2x cannot be equal to 0 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation. 2x 2 2x 5 2x 0 22x Subtract 2x from each side Simplify. The statement is not true, so the equation has no solution. b. 4(4x 2 5) 5 2(8x 2 0) Write original equation. 6x x 2 20 Distributive property Notice that the statement 6x x 2 20 is true for all values of x. So, the equation is an identity. Exercises for Example 3 7. x 7 5 0x (3x 2 2) 5 3(5x 2 ) 9. } 2 (6x 8) 5 3(x 3) 45