10 Real Numbers, Equations, and Inequalities

10.4 Solving Linear Inequalities Objectives Graph the solutions of inequalities on a number line. Use the addition property of inequality. Use the multiplication property of inequality. Solve inequalities using both properties of inequality. Use inequalities to solve application problems.

Graph Solutions of Inequalities Example 1 a) b) 3

Graph Solutions of Inequalities Example 2 4

Use the Addition Property of Inequality 5

Use the Addition Property of Inequality Example 3 Solve 5 + 6x ≤ 5x + 8, and graph the solution set. 5 + 6x ≤ 5x + 8 A graph of the solution set is –5x –5x 5 + x ≤ 8 ] –5 –5 x ≤ 3

Use the Multiplication Property of Inequality For any real numbers A, B, and C (C ≠ 0), 1. If C is positive, then the inequalities A < B and AC < BC have exactly the same solutions; 2. If C is negative, then the inequalities A < B and AC > BC have exactly the same solutions. In words, each side of an inequality may be multi- plied by the same positive number without changing the solutions. If the multiplier is negative, we must reverse the direction of the inequality symbol.

Using the Multiplication Property of Inequality Example 4 Solve 6y > 12, and graph the solution set. 6y > 12 y > 2 A graph of the solution set is (

Use the Multiplication Property of Inequality Example 5 9

Solve Inequalities Using Both Properties of Inequality Solving Inequalities Step 1 Simplify each side separately. Use the distributive property to remove parentheses and combine terms. Step 2 Use the addition property. Add or subtract the same amount on both sides of the inequality so that the variable term ends up by itself on one side of the inequality sign and a number is by itself on the other side. You may have to have to do this step more than once. Step 3 Use the multiplication property to write the inequality in the form x < c or x > c. Reverse the direction of the inequality symbol when multiplying or dividing by a negative number.

Solve Inequalities Using Both Properties of Inequality Example 6 Solve –2(z + 3) – 5z ≤ 4(z – 1) + 9. Graph the solution set. –2(z + 3) – 5z ≤ 4(z – 1) + 9 A graph of the solution set is –2z – 6 – 5z ≤ 4z – 4 + 9 [ –7z – 6 ≤ 4z + 5 –4z –4z –11z – 6 ≤ 5 +6 + 6 Reverse the direction of the inequality symbol when dividing each side by a negative number. –11z ≤ 11 z ≥ –1

Use Inequalities to Solve Applied Problems Phrase Example Inequality Is greater than A number is greater than 4 x > 4 Is less than A number is less than –12 x < –12 Is at least A number is at least 6 x ≥ 6 Is at most A number is at most 8 x ≤ 8

Use Inequalities to Solve Applied Problems Example 8 Brent has test grades of 86, 88, and 78 on his first three tests in geometry. If he wants an average of at least 80 after his fourth test, what are the possible scores he can make on that test? Let x = Brent’s score on his fourth test. To find his average after four tests, add the test scores and divide the sum by 4. 252 + x ≥ 320 –252 –252 x ≥ 68 Brent must score 68 or more on the fourth test to have an average of at least 80.