402 CHAPTER 3 Polynomial and Rational Functions 3.6 Polynomial and Rational Inequalities ■ Polynomial Inequalities ■ Rational Inequalities Polynomial Inequalities Recall that a linear inequality is a first-degree inequality. Consider, for example, the linear inequality 2x - 1 65. Its solution set, 5x x 636, is found by isolating the variable using properties of inequality. Written in interval notation, the solution set is 1-∞, 32. A quadratic inequality such as 2x2 - 5x - 3 70 is a nonlinear inequality because it contains a variable term of degree other than 1. Both linear and quadratic inequalities are polynomial inequalities. Polynomial Inequality Let ƒ1x2 be a polynomial. A polynomial inequality is an inequality that can be written in the form ƒ1x2 60.* * The symbol 6can be replaced with 7, …, or Ú. Consider again the quadratic inequality 2x2 - 5x - 3 70. Using previous methods, we would solve the corresponding quadratic equation and identify the intervals that are solutions to the inequality using test values. 2x2 - 5x - 3 = 0 Corresponding quadratic equation 12x + 121x - 32 = 0 Factor. 2x + 1 = 0 or x - 3 = 0 Zero-factor property x = - 1 2 or x = 3 Solve each equation. The values -1 2 and 3 cause the expression 2x 2 - 5x - 3 to equal 0 and can be used to divide the number line into the intervals A -∞, - 1 2B, A - 1 2, 3B, and 13, ∞2. See Figure 59. Interval A Interval C (3, `) –1 0 4 3 ? Interval B (– , 3) 1 2 (–`, – ) 1 2 1 2 2(–1)2 – 5(–1) – 3 . 0 4 . 0 True Test Value ? 2(4)2 – 5(4) – 3 . 0 9 . 0 True Test Value ? 2(0)2 – 5(0) – 3 . 0 –3 . 0 False Test Value – Figure 59 Testing a value from each interval shows that the solution set is A -∞, - 1 2B ´ 13, ∞2.
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