Section 6.5 Solving Linear Inequalities Math in Our World

Learning Objectives  Graph solution sets for simple inequalities.  Solve linear inequalities in one variable.  Solve three-part linear inequalities.  Solve real-world problems using inequalities.

Linear Inequalities To solve a linear inequality means to find the set of all numbers that make the inequality a true statement when substituted in for the variable. That set is called the solution set for the inequality.

Graphing Inequalities

EXAMPLE 1 Graphing Solution Sets for Simple Inequalities Graph the solution set for each inequality. (a) x ≤ 10 (b) y > 4 (c) 30 < x ≤ 50

Solving Linear Inequalities If you multiply or divide both sides of an inequality by the same negative real number, the direction of the inequality symbol is reversed. If a < b and c < 0 then ac > bc and ac bc, and likewise for >, ≤, ≥.

EXAMPLE 2 Solving a Linear Inequality Solve and graph the solution set for 5x – 9 ≥ 21.

EXAMPLE 3 Solving a Linear Inequality Solve and graph the solution set for 16 – 3x > 40.

EXAMPLE 4 Solving a Linear Inequality Solve and graph the solution set for 4(x + 3) < 2x – 26.

EXAMPLE 5 Solving a Three-Part Linear Inequality Solve and graph the solution set for – 4 < 3 – 2y ≤ 9.

Common Phrases in Inequalities

EXAMPLE 6 Applying Inequalities to Vacation Planning With the stress of finals behind you, you decide to plan a vacation to relax a little bit. After poking around on the Internet, you find a room in the area you want to visit for $65 per night. Some quick estimating leads you to conclude that you’ll need at least $250 for gas, food, beverages, and entertainment expenses. Upon checking your bank balance, you decide that you can afford to spend at most $600 on the trip. How many nights can you stay?

EXAMPLE 7 Applying Inequalities to the Cost of Buying Food Mike is planning to buy lunch for himself and some coworkers. He decides to buy cheeseburgers and fries from the value menu—the burgers are $1 each, and the fries cost $0.80. He also needs to pay 5% of the total in sales tax. What is the largest number of items he can buy if he wants to buy the same number of burgers as fries, and he only has $10 to spend?

Classwork p : 5-53 eoo, odd