Copyright © 2013 Pearson Education, Inc. Section 2.5 Linear Inequalities

Solutions and Number Line Graphs A linear inequality results whenever the equals sign in a linear equation is replaced with any one of the symbols, or ≥. x > 5, 3x + 4 < 0, 1 – y ≥ 9 A solution to an inequality is a value of the variable that makes the statement true. The set of all solutions is called the solution set. Page 136

Example Use a number line to graph the solution set to each inequality. a. b. c. d. Page 137

Linear Inequalities in One Variable ) [ ]( Page 137

Interval Notation Each number line graphed on the previous slide represents an interval of real numbers that corresponds to the solution set to an inequality. Brackets and parentheses can be used to represent the interval. For example: Page 137

Example Interval Notation Write the solution set to each inequality in interval notation. a. b. Solution a.b. More examples Page 137

Example Checking a Solution Determine whether the given value of x is a solution to the inequality. Solution Page 138

The Addition Property of Inequalities Page 139

Example Solve each inequality. Then graph the solution set. a. x – 2 > 3b x ≤ 6 + x Solution a. x – 2 > 3 x – > x > 5 b x ≤ 6 + x 4 + 2x – x ≤ 6 + x – x 4 + x ≤ 6 4 – 4 + x ≤ 6 – 4 x ≤ 2 Page 139

Properties of Inequalities ) [ Page 140

The Multiplication Property of Inequalities Page 141

Example Solve each inequality. Then graph the solution set. a. 4x > 12b. Solution a. 4x > 12 b. Page 141

Example Solve each inequality. Write the solution set in set-builder notation. a. 4x – 8 > 12b. Solution a. 4x – 8 > 12 b. Page 142

Properties of Inequalities ) Page 142 ( Sign changes

Linear Inequalities [ Add 2 Add 3x same Page 142 Subtract 6 Subtract 5x Sign changes

Linear Inequalities Add 8 Sub 2x [ Page 142

Applications To solve applications involving inequalities, we often have to translate words to mathematical statements. Page 143

Example Translate each phrase to an inequality. Let the variable be x. a. A number that is more than 25. b. A height that is at least 42 inches. x > 25 x ≥ 42 Page 143

Example For a snack food company, the cost to produce one case of snacks is $135 plus a one-time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. a. Write a formula that gives the cost C of producing x cases of snacks. b. Write a formula that gives the revenue R from selling x cases of snacks. C = 135x + 175,000 R = 250x Page 144

Example (cont) For a snack food company, the cost to produce one case of snacks is $135 plus a one- time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. c. Profit equals revenue minus cost. Write a formula that calculates the profit P from selling x cases of snacks. P = R – C = 250x – (135x + 175,000) = 115x – 175,000 Page 144

Example (cont) For a snack food company, the cost to produce one case of snacks is $135 plus a one- time fixed cost of $175,000 for research and development. The revenue received from selling one case of snacks is $250. d. How many cases need to be sold to yield a positive profit? 115x – 175,000 > 0 115x > 175,000 x > Must sell at least 1522 cases. Page 144

DONE

Objectives Solutions and Number Line Graphs The Addition Property of Inequalities The Multiplication Property of Inequalities Applications

EXAMPLE Graphing inequalities on a number line Use a number line to graph the solution set to each inequality. a. b. c. d.