406 CHAPTER 3 Polynomial and Rational Functions EXAMPLE 4 Solving a Rational Inequality Using a Graph Solve x - 5 x + 2 Ú 0. SOLUTION Step 1 The inequality is already written with 0 on one side. Step 2 Let ƒ1x2 = x - 5 x + 2 and graph. The vertical asymptote has equation x = -2, and the horizontal asymptote has equation y = 1. The x-intercept, found by setting the numerator equal to 0, is 15, 02. Evaluating ƒ102 gives the y-intercept A0, - 5 2B. The graph does not intersect its horizontal asymptote because ƒ1x2 = 1 has no solution. 1 = x - 5 x + 2 Set ƒ1x2 = 1. x + 2 = x - 5 Multiply each side by x + 2. 2 = -5 Subtract x. The graph is shown in Figure 65. Step 3 The solution set of the inequality x - 5 x + 2 Ú 0 includes the x-values for which the graph of ƒ1x2 in Figure 65 lies on or above the x-axis. Because the inequality is nonstrict, the zero of ƒ1x2—here 5—is included in the solution set 1-∞, -22 ´ 35, ∞2. S Now Try Exercise 53. –8 –4 4 8 –8 –4 4 8 x y y = 1 x = –2 f(x) = x – 5 x + 2 (5, 0) 0 Figure 65 A false statement results. CAUTION When solving a rational inequality, remember that a value that causes any denominator to equal 0 must be excluded from the solution set. To be included in the solution set, a value must first be included in the domain of the expression that defines the rational inequality. EXAMPLE 5 Solving a Rational Inequality Using a Graph Solve 2 x + 3 6 1 x - 1 . SOLUTION Step 1 2 x + 3 - 1 x - 1 60 Subtract 1 x - 1 so that 0 is on one side. 21x - 12 1x + 321x - 12 - 11x + 32 1x - 121x + 32 60 Use 1x + 321x - 12 as the common denominator. 21x - 12 - 11x + 32 1x + 321x - 12 60 Write as a single fraction. 2x - 2 - x - 3 1x + 321x - 12 60 Distributive property x - 5 1x + 321x - 12 60 Combine like terms in the numerator. Note the careful use of parentheses. Rational Inequalities We use the same steps to solve rational inequalities using graphing as when solving polynomial inequalities.
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