Sullivan Algebra and Trigonometry: Section 1.5 Solving Inequalities Objectives of this Section Use Interval Notation Use Properties of Inequalities Solve Inequalities Solve Combined Inequalities

A closed interval denoted by [a, b], consists of all real numbers x for which a < x < b. [] ab An open interval, denoted (a, b), consists of all real numbers x for which a < x < b. () ab A half-open, or half-closed interval is [a, b), consisting of real numbers x for which a < x < b. [) ab

[ a ] a

Write the inequality -3 < x < 2 using interval notation. Illustrate the inequality using a real number line. [) -3 2

Properties of Inequalities Nonnegative Property Addition Property of Inequalities If a < b, then a + c < b + c. If a > b, then a + c > b + c. Example: Since - 4 < 5, then < 5 + 2

Multiplication Property of Inequalities If a 0, then ac < bc. If a bc. If a > b and if c > 0, then ac > bc. If a > b and if c < 0, then ac < bc.

Reciprocal Property for Inequalities

Solving Inequalities Procedures That Leave The Inequality Unchanged 1. Simplify both sides of the inequality by combining like terms and eliminating parenthesis. 2. Add or subtract the same expression on both sides of the inequality. 3. Multiply or divide both sides of the inequality by the same positive number.

Procedures That Reverse The Direction of the Inequality 1. Interchange the two sides of the inequality. 2. Multiply or divide both sides of the inequality by the same negative expression. Example: Solve -2x < 8 x > - 4

Solution set:   xx  33 or,

Solution set:    xx  2 or-2, -301 [

Solution set:   xx  3232 or, ()