Intermediate Algebra (11th edition)

552 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

OBJECTIVES

Polynomial and Rational Inequalities

9.7

1 Solve quadratic inequalities. 2 Solve polynomial inequalities of degree 3 or greater. 3 Solve rational inequalities.

OBJECTIVE 1 Solve quadratic inequalities.Now we combine the methods of

solving linear inequalities with the methods of solving quadratic equations to solve

quadratic inequalities.

Quadratic Inequality

A quadratic inequalitycan be written in the form

,

, or

where a, b, and care real numbers, with aZ 0.

ax^2 bxc◊ 0 ax^2 bxc»0,

ax^2 bxc< 0 ax^2 bxc>0,

One way to solve a quadratic inequality is by graphing the related quadratic function.

Solving Quadratic Inequalities by Graphing

Solve each inequality.

(a)

To solve the inequality, we graph the related quadratic function defined by

We are particularly interested in the x-intercepts, which are found as in

Section 9.6by letting and solving the following quadratic equation.

Factor.

or Zero-factor property

or The x-intercepts are and

The graph, which opens up since the coefficient of is positive, is shown in

FIGURE 18(a). Notice from this graph that x-values less than or greater than 4 result

in y-values greater than0. Thus, the solution set of written in

interval notation, is 1 - q, - 32 ́ 1 4, q 2.

x^2 - x- 127 0,

- 3

x^2

x= 4 x= - 3 1 4, 0 2 1 - 3, 0 2.

x- 4 = 0 x+ 3 = 0

1 x- 421 x+ 32 = 0

x^2 - x- 12 = 0

ƒ 1 x 2 = 0

x^2 - x- 12.

ƒ 1 x 2 =

x^2 - x- 1270

EXAMPLE 1

x

y

The graph is above the x-axis for (–, –3) (4, ).

–12

0

y > 0 –3 4

y > 0

x-values for which y > 0

f(x) = x^2 – x – 12

(a) (b) FIGURE 18

x

y