180 CHAPTER 3 Linear and Quadratic Functions 1 Solve Inequalities Involving a Quadratic Function In this section we solve inequalities that involve quadratic functions. • To solve the inequality ax bx c a 0 0 2 + + > ≠ graph the quadratic function f x ax bx c, 2 ( ) = + + and, from the graph, determine where the function is above the x-axis—that is, where f x 0. ( ) > • To solve the inequality ax bx c a 0 0 2 + + < ≠ graph the quadratic function f x ax bx c, 2 ( ) = + + and, from the graph, determine where the function is below the x-axis—that is, where f x 0. ( ) < • If the inequality is not strict, include the x-intercepts, if any, in the solution. Now Work PROBLEM 9 Solving an Inequality Solve the inequality x x4 12 0 2 − − ≤ and graph the solution set. EXAMPLE 1 Figure 35 Y x x4 12 1 2 = − − 8 220 24 8 Figure 34 f x x x4 12 2 ( ) = − − x y –4 4 8 4 8 –4 –8 –12 –16 (–2, 0) (6, 0) (0, –12) (2, –16) By Hand Solution Graph the function f x x x4 12. 2 ( ) = − − • y-intercept: f 0 12 ( ) = − Evaluate f at 0. • x-intercepts (if any): x x4 12 0 2 − − = Solve f x 0. ( ) = x x x x 6 2 0 6 0 or 2 0 ( )( ) − + = − = + = Factor. Use the Zero-Product Property. x 6 or = x 2 = − The y-intercept is 12; − the x-intercepts are 2− and 6. The vertex is at x b a2 4 2 1 2. = − = − − ⋅ = Because f 2 16, ( ) = − the vertex is 2, 16 . ( ) − See Figure 34 for the graph. Graphing Utility Solution Graph Y x x4 12. 1 2 = − − Figure 35 shows the graph using a TI-84 Plus CE. Use the ZERO command to find that the x-intercepts of Y1 are 2− and 6. Because we are solving f x 0, ( ) ≤ we must find the x-values for which the graph is below the x-axis, which is between 2− and 6. Since the inequality is not strict, the solution set is x x 2 6 { } − ≤ ≤ or, using interval notation, 2, 6 . [ ] − Because we are solving f x 0, ( ) ≤ we must find the x-values for which the graph is below the x-axis, which is between 2− and 6. Since the original inequality is not strict, include the x-intercepts. The solution set is x x 2 6 { } − ≤ ≤ or, using interval notation, 2, 6 . [ ] − See Figure 36 for the graph of the solution set. Figure 36 x 24 22 0 2 4 6 8 Solving an Inequality Solve the inequality x x 2 10 2 < + and graph the solution set. EXAMPLE 2 Solution Option 1 Rearrange the inequality so that 0 is on the right side. x x x x 2 10 2 10 0 2 2 < + − − < Subtract x 10 + from both sides. This inequality is equivalent to the original inequality.
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