SECTION 4.7 Polynomial and Rational Inequalities 259 In this section we solve inequalities that involve polynomials of degree 3 and higher, along with inequalities that involve rational functions. To help understand the algebraic procedure for solving such inequalities, we use the information obtained in Sections 4.1–4.3 and 4.5 about the graphs of polynomial and rational functions. The approach follows the same logic used to solve inequalities involving quadratic functions. 1 Solve Polynomial Inequalities Graphically and Algebraically OBJECTIVES 1 Solve Polynomial Inequalities Graphically and Algebraically (p. 259) 2 Solve Rational Inequalities Graphically and Algebraically (p. 261) 4.7 Polynomial and Rational Inequalities Now Work the ‘Are You Prepared?’ problems on page 263. • Solving Linear Inequalities (Section A.9, pp. A80–A82) • Solving Quadratic Inequalities (Section 3.5, pp. 180–181) PREPARING FOR THIS SECTION Before getting started, review the following: Solving a Polynomial Inequality Using Its Graph Solve ( )( ) + − > x x 3 1 0 2 by graphing ( ) ( )( ) = + − f x x x 3 1 2 . EXAMPLE 1 Figure 57 ( ) ( )( ) = + − f x x x 3 1 2 26 x y 22 24 2 3 23 6 9 12 4 (23, 0) (21.67, 9.48) (1, 0) End behavior y = x3 End behavior y = x3 By Hand Graphical Solution Graph ( ) ( )( ) = + − f x x x 3 1 2 and determine the intervals of x for which the graph is above the x -axis. Do you see why? These values of x result in ( ) f x being positive. Using Steps 1 through 8 on page 210, we obtain the graph shown in Figure 57. Graphing Utility Solution Graph ( )( ) = + − Y x x 3 1 1 2 . See Figure 58 using a TI-84 Plus CE. Using the ZERO command, find that the x -intercepts of the graph of Y1 are −3 and 1. The graph of Y1 is above the x -axis (and therefore f is positive) for − < < x 3 1 or > x 1.Therefore, the solution set is { } − < < > x x x 3 1 or 1 or, using interval notation, ( ) ( ) − ∪ ∞ 3, 1 1, . From the graph, we can see that ( ) > f x 0 for − < < x 3 1 or > x 1. The solution set is { } − < < > x x x 3 1 or 1 or, using interval notation, ( ) ( ) − ∪ ∞ 3, 1 1, . Figure 58 ( )( ) = + − Y x x 3 1 1 2 12 24 24 3 Now Work PROBLEM 9
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