Algebra 1 Keystone Remediation Packet Module 1 Anchor 3

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1 Algebra 1 Keystone Remediation Packet Module 1 Anchor 3 A Write, solve, and/or graph linear inequalities using various methods. A Write or solve compound inequalities and/or graph their solution sets on a number line. A Identify or graph the solution set to a linear inequality on a number line A Interpret solutions to problems in the context of the problem situation A Write, solve, and/or graph systems of linear inequalities using various methods (including graphing). Interpret solutions in the context of the problem Name

2 Inequalities Inequality-a mathematical sentence that compares expressions. An inequality contains the symbol <, >,, JK. The graph of an inequality shows the solution set of the inequality on a number line. An open circle is used when a number is not a solution. A closed circle is used when a number is a solution. An arrow shows that the graph continues in that direction. Example 1: The graphs show the height restrictions h (in inches) for two rides at an amusement park. Write an inequality that represents the height restrictions for each ride. Answer: Rider A S 48 A rider may be 48 inches or taller. Rider B S < 52 A rider must be less than 52 inches. Example 2: The graph of a compound inequality with and is the intersection of the graphs of the inequalities. The graph shows numbers that are solutions of both inequalities. 2 S < 5 means S 2 XYZ S < 5 Example 3: The graph of a compound inequality with or is the union of the graphs of the inequalities. The graph shows numbers that are solutions of either inequality. [ 2 JK [ > 1

3 Solving One Step Inequalities Just as you used inverse operations to produce equivalent equations, you can use the properties of inequalities to produce equivalent inequalities. Addition and subtraction examples: Solve and graph the linear inequality. a. S 6 10 b. b Multiplication and division with positive coefficients examples: c. 4d 36 d. e f > 5 Multiplication and division with negative coefficients examples: When you multiply or divide by a negative coefficient, the direction of the inequality symbol must be reversed to produce an equivalent inequality. e. 7[ 35 f. e hi 5

4 Solving Multi Step and Compound Inequalities a. Solve and graph 2k 4 8 b. Solve and graph 6S 5 < 2S + 11 c. Solve and graph 4 < 2S 2 < 4 d. Solve and graph 3 < 2S e. 3[ 5 < 8 JK 2[ 1 > 5

5 Solving Absolute Value Inequalities An absolute value inequality is an inequality that contains an absolute value expression. Rewrite the absolute value inequality as two inequalities and then solve. a. Solve 2S 1 11 b. Solve 2S 1 5

6 Keystone Practice Problems A Interpret solutions to problems in the context of the problem situation 1) Nitrogen is a gas at certain temperatures under normal conditions. These temperatures (t) in degrees Celsius ( ) are represented by the inequality o 196. Which best describes the meaning of the inequality? a. Nitrogen is a gas only if the temperature is below 196. b. Nitrogen is a gas only if the temperature is above 196. c. Nitrogen is a gas only if the temperature is at or below 196. d. Nitrogen is a gas only if the temperature is at or above ) A t-shirt company has a goal to earn a monthly profit of more than $3500. The company charges $20 per t-shirt The company has $1500 in monthly costs. 20S 1500 > 3500 The inequality represents this situation. Which best describes the meaning of x in this inequality? a. the profit made from the sale of 20 t-shirts b. the profit made from 1 month of t-shirt sales c. the number of t-shirts that need to be sold for the company to meet its goal d. the number of t-shirts that need to be sold for the company to recover its monthly costs 3) A cleaning service offers its customers a choice between two plans Plan A costs $3000 and includes 1 year of unlimited number of cleanings Plan B costs $75 per cleaning Henry wants to choose the less expensive plan. He uses the inequality 3000 < 75p to decide which plan to choose based on the number of cleanings p, he expects to need. Based on the solution of the inequality, which statement about Henry s choice of plan is true? a. Henry should choose Plan A only if he expects to need fewer than 40 cleanings in 1 year. b. Henry should choose Plan A only if he expects to need more than 40 cleanings in 1 year. c. Henry can choose either plan and pay the same amount if he expects to need fewer than 40 cleanings in 1 year. d. Henry can choose either plan and pay the same amount if he expects to need more than 40 cleanings in 1 year.

7 4) A baseball team had $1000 to spend on supplies. The team spent $185 on a new bat. New baseballs cost $4 each. The inequality d 1000 can be used to determine the number of new baseballs (b) that the team can purchase. Which statement is true? a. The team can purchase 204 new baseballs. b. The minimum number of new baseballs that can be purchased is 185. c. The maximum number of new baseballs that can be purchased is 185. d. The team can purchase 185 new baseballs, but this number is neither the maximum nor the minimum. 5) The science department at a high school has $1,500 to spend on supplies. They spend $385 on chemicals for the science class. Dissection frogs cost $6 each. The inequality s 1500 can be used to determine the number of frogs that the department can purchase. Which statement is true? a) The department can purchase 250 frogs. b) The maximum number of frogs that can be purchased is 185. c) The minimum number of frogs that can be purchased is 185. d) The department can purchase 185 frogs, but that is neither the maximum or the minimum. 6) Paul has $1250 in his checking account. He only uses the account for paying his car insurance bill. The bank will notify Paul when his account balance is less than $400. Each month, his insurance bill is $95. The inequality d 1250 can be used to determine the number of monthly bills that Paul can make before the bank contacts him. Which statement about the number of payments is true? a) Paul can make a minimum of 13 payments before he is contacted. b) Paul can make a maximum of 9 payments before he is contacted. c) Paul can make a maximum of 8 payments before he is contacted. d) Paul can make a minimum of 4 payments before he is contacted. 7) A ticket to a baseball game costs $20. Each soda sold at the game costs $5. Shawn may spend no more than $50. He buys 1 ticket a S sodas. Shawn represents this situation with the inequality 5S The solution of the inequality is S 6. Which statement best describes the solution of the inequality? a. Shawn buys 6 or fewer sodas b. Shawn buys 6 or fewer tickets c. Shawn buys 1 ticket and 5 sodas. d. Shawn has less than or equal to $6 remaining when he leaves the game.

8 8) Dennis can travel 585 miles on a full tank of gas. After filling his tank, he traveled 195 miles. Each hour, Dennis travels 65 miles. The inequality h can be used to determine the number of hours (h) that Dennis can travel before running out of fuel. Which statement about the number of hours is true? a. Dennis can travel for 9 hours before running out of fuel. b. The maximum number of hours Dennis can drive is 6. c. The minimum number of hours Dennis can drive is 6. d. The minimum number of hours Dennis can drive is 9. 9) A football team has a budget of $650 for supplies. The team purchased a case of water bottles for $75. Each new football costs $40. The inequality s 650 can be used to determine the number of footballs that the team can purchase. Which statement about the number of footballs is true? a. The team can purchase 16 footballs. b. The minimum number of footballs that can be purchased is 14. c. The team can purchase 15 footballs. d. The maximum number of footballs that can be purchased is ) One of Kara s problems asks her to solve the inequality 3(3 S) 18 Kara s work is shown below. Rewrite the incorrect step so that it is correct and then calculate the correct answer.

9 Keystone Practice Problems A Identify or graph the solution set to a linear inequality on a number line 1) The solution set of an inequality is shown below. Which inequality has the solution set shown in the graph? a. he x b. he x e c. d. hy z y z hy x z e x y z 2) Patricia keeps apples in 3 bins and 2 crates in her store. Each bin can hold no more than 200 pounds. Each crate can hold no more than 50 pounds. Which graph represents all of the possible weights, in pounds, of apples Patricia can keep in her store. *3) The inequality 2 3S 14 is shown below. Which graph represents the inequality?

10 4) An inequality is shown below. Which graph shows the solution of the inequality? 4S + 2 < 2S + 9 5) Graph the solution set to the inequality 4S + 3 < 7S 9 on the number line below. 6) Graph the solution set to the inequality 4S 18 < 16 4S on the number line below.

11 Keystone Practice Problems A Write or solve compound inequalities and/or graph their solution sets on a number line, may include absolute value inequalities) 1) Jason decided that he will sell his stocks if their value per share (x) goes below $5 of above $15. Which compound inequality represents the value at which Jason will sell his stocks? a) S > $5 JK S < $15 b) S < $5 JK S > $15 c) S > $5 XYZ S < $15 d) S < $5 XYZ S > $15 2) A compound inequality is shown below. Which graph represents the solution of the inequality? S 3 XYZ S 0 *3) What is the solution of this compound inequality? 5 < 2 3[ < 14? A. 4 > [ > 1 B. 4 < [ < 1 C. 1 > [ > 4 D. 1 < [ < 4

12 4) Which is a graph of the solution of the inequality 2S 1 5? 5) Which is a graph of the solution of the inequality 2S 7 3?

13 Systems of Inequalities A Write, solve, and/or graph systems of linear inequalities using various methods (including graphing). Interpret solutions in the context of the problem A system of linear inequalities is a set of two or more linear inequalities in the same variables. The graph of a system of linear inequalities is the graph of all the solutions of the system. Example: The graph below shows the solution set for the inequalities [ < 3 and [ < S + 2 You try: Write the system of inequalities that represents the solution set below:

14 Keystone Practice Problems 1) 2) 3) A group of friends will buy at most 8 snacks at a movie theater and spend no more than $42. They will pay $4 for each box of candy and $7 for each bag of popcorn. The system of inequalities graphed below represents this information. What combination of boxes of candy and bags of popcorn could the group buy?

15 4) Nate will buy at least 10 cans of fishing bait. Each large can costs $7, and each small can costs $3. He will spend up to $54 on bait. This system of inequalities is graphed below. Which combination of cans could Nate buy? a. 2 large cans and 6 small cans b. 2 large cans and 10 small cans c. 6 large cans and 10 small cans d. 8 large cans and 2 small cans 5) Kelly is planning a business trip to Hartford, CT and Boston, MA. She is going to stay no more than 5 nights on the trip. She estimates each night she stays at a hotel in Hartford will cost $90 and each night she stays at a hotel in Boston will cost $120. She wants to spend no more than $540 for hotel stays. Which graph represents the possible combination of nights she can spend in Hartford (x) and Boston (y)?

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