Name Class Date. Properties of Inequality

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1 Name Class Date 2-2 Solving Inequalities by Adding or Subtracting Going Deeper Essential question: How can you use properties to justify solutions to inequalities that involve addition and subtraction? A-REI.2.3 ENGAGE Properties of Inequality You have solved addition and subtraction equations by performing inverse operations that isolate the variable on one side. The value on the other side is the solution. Inequalities involving addition and subtraction can be solved similarly using the following inequality properties. These properties are also true for and. Addition Property of Inequality Subtraction Property of Inequality If a > b, then a + c > b + c. If a < b, then a + c < b + c. If a > b, then a - c > b - c. If a < b, then a - c < b - c. REFLECT a. How do the Addition and Subtraction Properties of Inequality compare to the Addition and Subtraction Properties of Equality? Most linear inequalities have infinitely many solutions. When using set notation, it is not possible to list all the solutions in braces. The solution x in set notation is {x x }. Read this as the set of all x such that x is less than or equal to. A number line graph can be used to represent the solution set of a linear inequality. the set of all x { x x } such that < or >, mark the endpoint with an empty circle. or, mark the endpoint with a solid circle. x is less than or equal to x < x x > x Chapter 2 79 Lesson 2

2 2 A-REI.2.3 EXAMPLE Adding to Find the Solution Set Solve. Write the solution using set notation. Graph your solution. A x - 3 < 2 x < 2 + Property of Inequality; add to both sides. x < B x x Property of Inequality; add to both sides. x REFLECT 2a. Is 5 in the solution set of the inequality in Part A? Explain. 2b. Suppose the inequality symbol in Part A had been >. Describe the solution set. 2c. Suppose the inequality symbol in Part B had been. Describe the solution set. Chapter 2 80 Lesson 2

3 3 A-REI.2.3 EXAMPLE Subtracting to Find the Solution Set Solve. Write the solution using set notation. Graph your solution. A x + 4 > 3 x > 3 - Property of Inequality x > B x x Property of Inequality x REFLECT 3a. Is -3 in the solution set of the inequality in Part B? Explain. 3b. Suppose the inequality symbol in Part A had been. Describe the solution set. 3c. Suppose the inequality symbol in Part B had been <. Describe the solution set. Chapter 2 8 Lesson 2

4 PRACTICE Solve. Justify your steps. Write the solution in set notation. Graph your solution.. x x - 2 > x + 6 < x + 3 < 2 5. x Chapter 2 82 Lesson 2

5 Name Class Date Additional Practice 2-2 Solve each inequality and graph the solutions.. b + 8 > 5 2. t x 4. g + 8 < m > d + 9 Answer each question. 7. Jessica makes overtime pay when she works more than 40 hours in a week. So far this week she has worked 29 hours. She will continue to work h hours this week. Write, solve, and graph an inequality to show the values of h that will allow Jessica to earn overtime pay. 8. Henry s MP3 player has 52MB of memory. He has already downloaded 287MB and will continue to download m more megabytes. Write and solve an inequality that shows how many more megabytes he can download. 9. Eleanor needs to read at least 97 pages of a book for homework. She has read 34 pages already. Write and solve an inequality that shows how many more pages p she must read. Chapter 2 83 Lesson 2

6 Problem Solving Write the correct answer.. Sumiko is allowed to watch no more than 0 hours of television each week. She has watched 4 hours of television already. Write and solve an inequality to show how many more hours of television Sumiko can watch. 2. A satellite will be released into an orbit of more than 400 miles above the Earth. The rocket carrying it is currently 255 miles above Earth. Write and solve an inequality to show how much higher the rocket must climb before it releases the satellite. 3. Wayne s homework is to solve at least 20 questions from his textbook. So far, he has completed 9 of them. Write, solve, and graph an inequality to show how many more problems Wayne must complete. 4. Felix wants to get at least one hour of exercise each day. Today, he has run for 40 minutes. Write, solve, and graph an inequality that shows how much longer Felix needs to exercise to reach his goal. The high school has been raising money for charity and the class that raises the most will be awarded a party at the end of the year. The table below shows how much money each class has raised so far. Use this information to answer questions The school has a goal of raising at least $3000. Which inequality shows how much more money m they need to raise to reach their goal? A m 25 C m 25 B m < 25 D m > The juniors would like to raise more money than the seniors. The seniors have completed their fundraising for the year. Which expression shows how much more money j the juniors must raise to overtake the seniors? F j 220 H j 220 G j < 220 J j > 220 Class Amount Raised ($) Seniors 870 Juniors 650 Sophomores 675 First-Years A local business has agreed to donate no more than half as much as the senior class raises. Which inequality shows how much money b the business will contribute? A 2 (870) b C (870) b 2 B b D b Chapter 2 84 Lesson 2