403 3.6 Polynomial and Rational Inequalities We can also solve the quadratic inequality 2x2 - 5x - 3 70 using graphing. The graph of the related quadratic function ƒ1x2 = 2x2 - 5x - 3 is shown in Figure 60. The solutions of ƒ1x2 = 0 correspond to the x-values of the x-intercepts of the graph, - 1 2 and 3. The solutions of ƒ1x2 70 are the x-values for which the graph lies above the x-axis. See the portions of the graph in blue. From the graph, we see that the solution set of 2x2 - 5x - 3 70 is A -∞, - 1 2B ´ 13, ∞2. The solutions of ƒ1x2 60 correspond to the x-values for which the graph lies below the x-axis. See the portion of the graph in red, which indicates that the solution set of the inequality 2x2 - 5x - 3 60 is A - 1 2, 3B. ƒ1x2 60 x y 9 (21, 4) (4, 9) (0, 23) f(x) 5 2x2 2 5x 2 3 4 26 3 4 0 1 2 – Figure 60 ƒ1x2 70 Solving a Polynomial Equation or Inequality Using a Graph Step 1 Rewrite the equation or inequality, if necessary, so that an expression is on one side with 0 on the other side. Step 2 Set the expression of the equation or inequality equal to ƒ1x2, and graph the related function. Step 3 Use the graph of ƒ1x2 to determine solutions as follows. • The real solutions of ƒ1x2 = 0 are the x-values of the x-intercepts of the graph. These are the zeros of ƒ1x2. • The real solutions of ƒ1x2 60 are the x-values for which the graph lies below the x-axis. • The real solutions of ƒ1x2 70 are the x-values for which the graph lies above the x-axis. EXAMPLE 1 Solving a Quadratic Equation and Inequalities Using a Graph Use the graph in Figure 61 to solve each equation or inequality. (a) -31x - 121x + 42 = 0 (b) -31x - 121x + 42 60 (c) -31x - 121x + 42 Ú 0 SOLUTION (a) The solutions of the equation -31x - 121x + 42 = 0 are the x-values of the x-intercepts of the graph of ƒ1x2, -4 and 1. See Figure 61. Therefore, the solution set is 5-4, 16. (b) The solution set of the inequality -31x - 121x + 42 60 includes the x-values for which the graph of ƒ1x2 lies below the x-axis. See the portions of the graph in red in Figure 61. The solution set is 1-∞, -42 ´ 11, ∞2. (c) The solution set of the inequality -31x - 121x + 42 Ú 0 includes the x-values for which the graph of ƒ1x2 lies on or above the x-axis. Because the inequality Ú is nonstrict, the zeros of the related graph are included. See the portion of the graph in blue in Figure 61, along with the x-values of the x-intercepts. The solution set is 3-4, 14. S Now Try Exercise 11. x y 20 f(x) 5 23(x 2 1)(x 1 4) 10 –5 2 –4 0 (24, 0) (1, 0) Figure 61 ƒ1x2 70 ƒ1x2 60
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