404 CHAPTER 3 Polynomial and Rational Functions In Example 1, we could have divided each equation and inequality by -3 to obtain equivalent forms. If this is done in part (b), for example, we must reverse the direction of the inequality symbol and then analyze the graph of the related function. -31x - 121x + 42 6 0 Quadratic inequality from Example 1(b) -31x - 121x + 42 -3 7 0 -3 Divide by -3. Reverse the direction of the inequality symbol. 1x - 121x + 42 7 0 Equivalent inequality The graph of the related function ƒ1x2 = 1x - 121x + 42 in Figure 62 gives the same solution set, 1-∞, -42 ´ 11, ∞2, as found in Example 1(b). Notice, however, that the graph of this parabola opens upward, and the solution set includes x-values for which the graph of ƒ1x2 lies above the x-axis. See the portions of the graph in blue. x y 6 f(x) 5 (x 2 1)(x 1 4) 3 –4 1 –4 –2 0 (24, 0) (1, 0) Figure 62 ƒ1x2 60 ƒ1x2 70 EXAMPLE 2 Solving a Polynomial Equation and Inequalities Using a Graph Solve each equation or inequality. (a) 51x - 1221x - 32 = 0 (b) 51x - 1221x - 32 60 (c) 51x - 1221x - 32 Ú 0 (d) 51x - 1221x - 32 … 0 SOLUTION Step 1 Each equation or inequality is written with an expression on one side and 0 on the other. Step 2 Let ƒ1x2 = 51x - 1221x - 32 and graph. The zeros of this function are 1 (multiplicity 2) and 3. Multiplying the factors and identifying the term of greatest degree show that the dominating term of ƒ1x2 is 5x3. Thus, the end behavior is . Evaluating ƒ102 gives the y-intercept 10, -152. Using the appropriate end behavior, draw a smooth curve that crosses the x-axis at 3 and touches the x-axis at 1, then turns and changes direction. The graph also passes through the y-intercept 10, -152. See Figure 63. Step 3 Use the graph in Figure 63 to determine the solution sets. (a) ƒ1x2 = 0 for the x-values of the x-intercepts of the graph of ƒ1x2 — that is, the zeros of ƒ1x2. The solution set of the equation 51x - 1221x - 32 = 0 is 51, 36. (b) ƒ1x2 60 —that is, the graph lies below the x-axis—for x 61 and 1 6x 63. Therefore, the solution set of the inequality 51x - 1221x - 32 60 is 1-∞, 12 ´ 11, 32. This is a strict inequality, so the zeros of ƒ1x2 are not included in the solution set. (c) ƒ1x2 Ú 0 —that is, the graph lies on or above the x-axis—for solution set 516´ 33, ∞2. (d) ƒ1x2 … 0 —that is, the graph lies on or below the x-axis—for solution set 1-∞, 34. S Now Try Exercise 23. x y f(x) 5 5(x 2 1)2(x 2 3) 10 –5 –15 1 3 –1 0 (1, 0) (3, 0) Figure 63 ƒ1x2 70 ƒ1x2 60
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