16. Two years of local Internet service costs $685, including the installation fee of $85. What is the monthly fee?

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1 Solving Two-Step Equations 11.1 Check each answer. 1. 7x y a a k m v u z f w e d u f Two years of local Internet service costs $685, including the installation fee of $85. What is the monthly fee? 11.1 CHALLENGE: Consecutive Number Search Consecutive numbers are numbers that come one after the other. For example, 3 and 4 are consecutive whole numbers. 6 and 8 are consecutive even numbers. 7 and 9 are consecutive odd numbers. If n a number, then n 1 is the next consecutive whole number after n. If n an even number, then n 2 is the next consecutive even number after n. If n an odd number, then n 2 is the next consecutive odd number after n. (Hint: Let n the first number. Then write an equation for each sum.) 1. The sum of two consecutive numbers is 73. What are the numbers? 2. The sum of two consecutive numbers is 131. What are the numbers? 3. The sum of two consecutive even numbers is 70. What are the numbers? 4. The sum of two consecutive even numbers is 158. What are the numbers? 5. The sum of two consecutive odd numbers is 108. What are the numbers? 6. The sum of two consecutive odd numbers is 180. What are the numbers?

2 Solving Multi-Step Equations x 8 3x n 3 4n h 6 7h g 6 2g b 7 3b y 1 3y k 14 3k m 10 14m d 18 4d (n 5) (v 6) (t 6) (m 3) (s 3 4 ) (c 4 5 ) Joel has twice as many CDs as Mariella has. Subtracting 7 from the number of CDs Joel has and dividing by 3 equals the number of CDs Blake has. If Blake has 25 CDs, how many CDs does Mariella have? 11.2 CHALLENGE: Solve the System When equations have two variables, you can sometimes find the point where two equations have the same solution that is, where one value of x and one value of y solve both equations. This is called solving a system of equations. To solve a system of equations, you can substitute from one equation into the other. Example: Solve x y 6 and x y 2. Step 1: Solve one equation for y. x y 6 y 6 x Step 2: Substitute that value of y into the second equation. Solve for x. x y 2 x (6 x) 2 x 6 x 2 2x 6 2 2x 8 x 4 Step 3: Substitute the solution for x from Step 2 into the other equation. x y 6 4 y 6 y 2 Step 4: Write the solution as an ordered pair (x, y). Solution: (4, 2) Solve each system of equations. 1. y 3x 2. x 4y 3. y x 2 4. x y 2 x y 20 x y 10 x y 12 3y x x y 3 6. y x 9 7. x 2 y 8. x y 2 x y 7 x y 3 2y x 9 3x y 8

3 Solving Equations with Variables on Both Sides 11.3 Group the terms with the variables on one side of the equal sign and simplify. Do not solve t 6t x 32 2x 3. j 20 4j 4. 5d 40 5d 5. 9m 28 2m x x 7. 8k 6k v 3v y 10 6y a a n n d 3 10 d Members of the Lake Shawnee Club pay $40 per summer season plus $7.50 each time they rent a boat. Nonmembers pay $12.50 each time they rent a boat. How many times would both a member and a nonmember have to rent a boat in order to pay the same amount? 11.3 CHALLENGE: Problem Solving 1. Five added to twice Erik s age is the same as 3 times his age minus 2. How old is Erik? 2. Three times the perimeter of a triangle is the same as 75 decreased by twice the perimeter. What is the perimeter of the triangle? 3. The area of a pentagon increased by 27 is the same as four times the area of the pentagon, minus 15. What is the area of the pentagon? 4. To repair body damage on a car, AutoBody charges $125, plus $18 per hour. CarCare charges $200, plus $12 per hour. Determine the number of hours for which the two body shops will cost the same. 5. Sandy and Suzanne are planting flower pots around the school building. Sandy has planted 33 pots and is planting at the rate of 10 pots per hour. Suzanne has planted 25 pots and is planting at the rate of 14 pots per hour. In how many hours will they have planted the same number of flower pots? 6. The length of the sides of a square measure 2x 5. The length of a rectangle measures 2x, and the width measures x 2. For what value of x is the perimeter of the square the same as the perimeter of the rectangle? 7. Louisa used Downtown Taxi, which charges $2 for the first mile and $1.10 for each additional mile. Pietro used Uptown Cab, which charges $5 for the first mile and $0.95 for each additional mile. They paid the same amount and traveled the same distance. How far did they travel? 8. Toni bought some beach towels on sale for $8 each. Theo bought the same number of beach towels at the full price of $12. Toni s total was $24 less than Theo s total. How many beach towels did they each buy?

4 Inequalities Write an inequality for each situation. 1. The temperature today will be at most 50 F. 2. The temperature tomorrow will be above 70 F. 3. Yesterday, there was less than 2 inches of rain. 4. Last Monday, there was at least 3 inches of rain Graph each inequality. 5. t 2 6. j > 5 7. y 0 8. b < 1 2 Graph each compound inequality. 9. f > 3 or f < w b < 0 or b y 3 or y < m < CHALLENGE: Square Equations Some equations can be solved using square roots. Example 1: Example 2: Solve x Solve w 2 8 w. Think: , and ( 5) First, cross multiply: w So, the equation has two solutions. Think: , and ( 4) x 5, 5 So, the equation has two solutions. y 4, 4 1. x y n a m s c 3 3 c t t 4 8. k 4 25 k d d 9 9. e 4 16 e 12. z p h v 2 110

5 Solving Inequalities by Adding or Subtracting 11.5 Then graph each solution set on a number line. 1. y n x h p s 7 16 Check each answer g w a z t b In order for a field trip to be scheduled, at least 30 students must sign up. So far, 23 students have signed up. At least how many more students must sign up in order for the field trip to be scheduled? 11.5 CHALLENGE: Absolute Value Equations The absolute value of a number x is: x x if x 0 and x x if x 0 When you solve an absolute value equation, you must consider both of these cases. Example: Solve n Case 1: n 8 20 Case 2: (n 8) 20 n 12 n 8 20 n 28 n 28 The equation has two solutions: n 12, t s y g m p c n e x k s w b r 1 17

6 Solving Inequalities by Multiplying or Dividing n b a t s r Check each answer. 7. 8c a t s b m It cost Sophia $530 to make wind chimes. How many wind chimes must she sell at $12 apiece to make a profit? 14. It cost the Wilson children $55 to make lemonade. How many glasses must they sell at 75 each to make a profit? 15. Jorge s soccer team is having its annual fund raiser. The team hopes to earn at least three times as much as it did last year. Last year the team earned $87. What is the team s goal for this year? 11.6 CHALLENGE: Problem Solving 1. U.S. Postal Service regulations state that a package can be mailed using Parcel Post rates if it weighs no more than 70 pounds. What is the maximum number of books weighing 12 pounds each that can be mailed in one box using Parcel Post rates? 2. Marc wants to buy a set of at least 6 antique chairs for his dining room. He has decided to spend no more than $390. What is the most he can spend per chair? 3. Alfonso earns $9.00 per hour working part-time as a lab technician. He wants to earn more than $144 this week. At least how many hours does Alfonso have to work? 4. Mrs. Menendez invited 8 children to her son s birthday party. She wants to make sure that each child gets at least 4 small prizes. At least how many prizes should she buy? 5. The Computer Club spent $2,565 on mouse pads. The members plan to sell the mouse pads during the book fair. If they charge $9.50 for each mouse pad, how many must they sell in order to make a profit? 6. The Parents Organization bought 1,000 bumper stickers at $1.25 each to sell at football games. They want to make at least $750 profit. What should be the selling price of the bumper stickers? 7. In 2000, the national ratio of students to computers with Internet access in public schools was 7:1. Winston School had 623 students. If the school had a lower ratio, how many computers with Internet access did Winston School have? 8. A new theme park averaged fewer than 2,000 visitors per week during the winter months. What was the average daily attendance?

7 Solving Multi-Step Inequalities 11.7 Then graph each solution set on a number line. 1. 5x r z (w 36) n 7 7n 5 6. u d (g 3) b ( a 8) c p 8 p Fifty students in the seventh grade are trying to raise at least $2,000 for sports supplies. They have already raised $750. How much should each student raise, on average, in order to meet the goal? 11.7 CHALLENGE: Problem Solving 1. Grace earns $7 for each car she washes. She always saves $25 of her weekly earnings. This week, she wants to have at least $65 in spending money. At least how many cars must she wash? 2. Monty has saved $400 to spend on a video game player and games. The player he wants costs $275. The games each cost $39. At most, how many games can he buy along with the player? 3. A video game club charges $8 per month as a membership fee, plus $2.75 for each game rental. Eugenie plans to join and rent no more than 5 games a month. What amount should she budget each month for video games? 4. Cooper Middle School has a goal of collecting more than 1,000 cans of food in a food drive. So far, 375 cans have been collected. During the last 13 days of the drive, at least how many cans must be collected each day in order to meet the goal?

Solving Equations with Variables on Both Sides

Practice A Solving Equations with Variables on Both Sides Group the terms with the variables on one side of the equal sign and simplify. Do not solve. 1. 9p 6p 21 2. 5t 14 2t 3. 2k 18 4k 4. 6u 48 6u 5.