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Avon High School Section: 1.7 ACE COLLEGE ALGEBRA II - NOTES Linear Inequalities and Absolute Value Inequalities Mr. Record: Room ALC-129 Day 1 of 1 Interval Notation and Set-Builder Notation Let a and b be real numbers such that a < b. Interval Notation Set-Builder Notation Graph a, b a, b a, b a, b a, a, ,b ,b , Solving Linear Inequalities in One Variable Example 1 a. 1,3 2,6 Finding Intersections and Unions of Intervals Use graphs to find each set. b. 1,3 2,6 Investigation: Consider the following true inequality statement: 2 < 5. Perform the following operations to both sides and determine if the statement remains true or becomes false. (i) Add 4 to both sides T F (ii) Subtract 1 from both sides T F (iii) Multiple both sides by 2 T F (iv) Multiply both sides by -3 T F (v) Divide both sides by -2 T F Example 2 Solving a Linear Inequality a. Solve 2 3x 5 b. Solve 3x 1 7 x 15 Investigation: For Example 2 – part b, use a graphing calculator to sketch each side of the inequality as an equation with y by itself. How could you determine the solution to the inequality by looking at the resulting graphs of the two equations? Example 3 Solving a Linear Inequality Containing Fractions Solve and graph the solution set on a number line: x4 x2 5 . 2 3 6 Inequalities With Unusual Solution Sets Example 4 a. Solving Linear Inequalities Solve each inequality. 3( x 1) 3 x 2 b. x 1 x 1 . Solving Compound Inequalities Example 5 Solving a Compound Inequality Solve and graph the solution set of the following inequality on a number line. 1 2x 3 11 This inequality could also be rewritten as: Solving Inequalities with Absolute Value Solving an Absolute Value Inequality If u is an algebraic expression and c is a positive number, 1. The solutions of u c are the real numbers that satisfy c u c. 2. The solutions of u c are the real numbers that satisfy u c or u c. To better understand these rules, stop and think about what the solution to |x| < 2 is. These rules are valid if < is replaced by ≤ and > is replaced by ≥. Example 6 Solving an Absolute Value Inequality Solve and graph the solution set of each inequality on a number line. a. x2 5 c. 18 6 3x b. 3 5x 2 20 19 Applications Example 7 Selecting the Better Deal Rent-A-Heap car rental agency charges $4 a day plus $0.15 per mile, whereas Clunkers-R-Us rental agency charges $20 a day and $0.05 per mile. How many miles must be driven to make the daily cost of a Rent-A-Heap rental a better deal that a Clunkers-R-Us rental?