1.6Solving Inequalities.

Trichotomy Property

a < b a = b 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 < b and if c > 0, then ac < bc.

If a < b and if c < 0, then ac > 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.

[ ]a b

An open interval, denoted (a, b), consists of all real numbers x for which a < x < b.

( ) a b

A half-open, or half-closed interval is (a, b], consisting of all real numbers x for which a < x < b.

( ]a b

A half-open, or half-closed interval is [a, b), consisting of all real numbers x for which a < x < b.

[ )a b

[a

(a

[ )−3 2,

[ )-3 20

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 Y1 and Y2 on the same screen.

• If inequality is of the form Y1 < Y2, determine on what interval Y1 is below Y2.

• Similarly for Y1 > Y2.

• If inequality is not strict include the endpoints of the intervals.

Y1<Y2, Y1>Y2, Y1≤Y2, Y1≥Y2

Inequalities Involving Absolute Value

3 1 5x − <

− < − <5 3 1 5x

− < <4 3 6x

− < <43

2x

Solution set: (-4/3, 2)

-2 -1 0 1 2 3 ( )

Solve absolute value inequality

Inequalities Involving Absolute Value

Solve Inequality