Chapter 6: Solving Linear Inequalities. Chapter 6.1: Solving Inequalities by Addition and Subtraction Goals for this section

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1 Chapter 6: Solving Linear Inequalities Chapter 6.1: Solving Inequalities by Addition and Subtraction Solving Inequalities by Addition: Example: Solve s 12 > 65. How to check solutions: How to graph a solution: Example: Solve 22 > m 8. Check your answer and graph it. Example: Solve d Check your answer and graph it. Solving Inequalities by Subtraction Example: By 5:00 PM the temperature in Fairbanks had risen 23 degrees to a temperature that is now less than 14 F. What was the temperature at the beginning of the day? Solve t + 23 < 14. Then graph the solution.

2 Example: The temperature t in a swimming pool increased 4 F since this morning. The temperature is now less than 81 F. What was the temperature this morning? Solve t + 4 < 81. Then graph the solution. Example: Solve 12n 4 13n. Graph the solution. Example: Solve 5h h. Graph the solution. Phrases that indicate inequalities in word problems: Example: Alicia wants to buy season passes to two theme parks. If one season pass costs $54.99 and Alicia has $100 to spend on both passes, the second season pass must cost no more than what amount? Chapter 6.2: Solving Inequalities by Multiplication and Division Solving Inequalities by Multiplication Example: Bob walks at a rate of ¾ mile per hour. He knows that it is at least 9 miles to Onyx Lake. How long will it take Bob to get there? Write and solve an inequality to find the length of time.

3 Example: Solve d 6 n p > 10 3 Solving Inequalities by Division: Example: Solve 12s 60 9r < 27. 8q < g 40. Chapter 6.3: Solving Multi-Step Inequalities Example: ABC Cellphones advertises a plan with 400 minutes per month for less than the competition. The price includes the $3.50 local tax. If the competition charges $43.50, what does ABC Cellphones charge for each minute?

4 Example: The inequality F > 212 represents the temperature in degrees Fahrenheit for which water is a gas (steam). The inequality C + 32 > 212 represents the temperature in degrees Celsius for which water is a gas. Find the temperatures in degrees Celsius for which water is a gas. Example: Solve 13 11d w 43 > 4y + 11 Example: Define a variable, write an inequality, and solve the problem. Check your solution. Four times a number plus twelve is less than a number minus three. Example: Solve 6c + 3(2 c) 2c + 1 6(5z 3) 36z 2(h + 6) > 3(8 h) Results: Example: Solve 7(s + 4) + 11s 8s 2(2s + 1) 18 3(8c + 4) 6(4c 1) 46 8m 4(2m + 5)

5 Chapter 6.4: Solving Compound Inequalities And Inequalities: Example: Graph the solution set of y 5 and y < 12. Example: Graph the solution set of a > 5 and a < 0. Example: Solve 7 < z Graph the solution set. Example: Solve y 3 11 and y 3 8. Graph the solution set. Or Inequalities: Symbols Intersection: Union: Example: A ski resort has several types of hotel rooms and cabins. The hotel rooms cost at most $89 per night, and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount a guest would pay per night at the resort.

6 Example: A store is offering a $30 mail-in rebate on all color printers. Luisana is looking at different color printers that range in price from $175 to $260. How much can she expect to spend after the mailin rebate? Example: Solve 4k 7 25 or 12 9k 30. Graph the solution set. Example: Solve x 9 or 2 + 4x < 10. Graph the solution set. Chapter 6.5: Solving Open Sentences Involving Absolute Value Absolute Values: Absolute Value Equations: When solving equations that involve absolute value, there are two cases to consider. Case 1: Case 2: Example: Solve 2x 1 = 7.

7 Example: Solve y + 2 = 4. Then graph the solution set. Example: Solve 3n 4 = 1. Then graph the solution set. Example: Write an open sentence involving absolute value for the graph. Graphing Absolute Value Functions: Example: Graph f(x) = x + 3

8 Example: Graph f(x) = x 5 Example: Graph g(x) = x Chapter 6.6: Solving Inequalities Involving Absolute Value Absolute Value Inequalities: Case 1: Case 2:

9 Example: Solve each open sentence. Then graph the solution set. s 3 12 n 8 2 x + 6 < 8 Absolute Value Inequalities: Case 1: Case 2: Example: Solve each open sentence. Then graph the solution set. 3y 3 > 9 2k + 1 > 7 2x Concept Summary: Example: The average annual rainfall in California for the last 100 years is 23 inches. However, the annual rainfall can differ by 10 inches from the 100 year average. What is the range of the annual rainfall for California? Example: The melting point of ice is 0 Celsius. During a chemistry experiment, Jill observed ice melting within 2 degrees. Write the range of temperatures that Jill observed ice melting.

10 Chapter 6.7: Graphing Inequalities in Two Variables Example: Graph 2y 4x > 6. Steps for Graphing a Linear Inequality Example: Graph x 1. Example: Graph y > x + 3 Example: Lee Cooper writes and edits short articles for a local newspaper. It takes her about an hour to write an article and about a half-hour to edit an article. If Lee works up to 8 hours a day, how many articles can she write in one day?

11 Chapter 6.8: Graphing Systems of Inequalities Example: Solve the system of inequalities by graphing. y < 2x + 2 y x 3 y 3 x + y 1 y 3x + 1 y 3x 2 2x + y 2 2x + y < 4

12 Example: A college service organization requires that its members maintain at least a 3.0 grade point average and volunteer at least 10 hours a week. Graph the requirements. Example: The LDL or bad cholesterol of a teenager should be less than 110. The HDL or good cholesterol of a teenager should be between 35 and 59. Make a graph showing appropriate levels of cholesterol or a teenager.