1.6 Solving Inequalities.

Trichotomy Property a b

Nonnegative Property

Transitive Property of Inequalities If a < b and b < c,then a < c. If a > b and b > c, then a > c.

Addition Property of Inequalities If a < b, then a + c < b + c. If a > b, then a + c > b + c.

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.

3 < 5 2(3) < 2(5) 3 < 5 -2(3) > -2(5) -6 > -10

Reciprocal Property for 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 all real numbers x for which a < x < b. (] ab

A half-open, or half-closed interval is [a, b), consisting of all 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.

Steps for Solving Inequalities Graphically Write the inequality in one of the following forms: Graph Y 1 and Y 2 on the same screen. If inequality is of the form Y 1 < Y 2, determine on what interval Y 1 is below Y 2. Similarly for Y 1 > Y 2. If inequality is not strict include the endpoints of the intervals.

Inequalities Involving Absolute Value

Solution set: (-4/3, 2) ( ) Solve absolute value inequality

Inequalities Involving Absolute Value

Solve Inequality