4 Solve an Inequality of Higher Degree Other polynomial inequalities in factored form can be solved in the same way that we solve quadratic inequalities. EXAMPLE 4 Solve (c 2)(c 5)(c 4) 0. Solution This is the factored form of a third-degree polynomial. Since the inequality is , the solution set will contain the intervals where (c 2)(c 5)(c 4) is negative. Solve (c 2)(c 5)(c 4) 0. c 2 0 or c 5 0 or c 4 0 Set each factor equal to 0. c 2 or c 5 or c 4 Solve. Put c 2, c 5, and c 4 on a number line, and test a number in each interval. Interval c 5 5 c 2 2 c 4 c 4 Test number c 6 c 0 c 3 c 5 Evaluate (6 2)(6 5)(6 4) (0 2)(0 5)(0 4) (3 2)(3 5)(3 4) (5 2)(5 5)(5 4) (c 2)(c 5)(c 4) (8)(1)(10) (2)(5)(4) (1)(8)(1) (3)(10)(1) 80 40 8 30 Sign Negative Positive Negative Positive Negative Positive Negative Positive 5 2 4 (c 2) (c 5) (c 4) We can see that the intervals where (c 2)(c 5)(c 4) is negative are (q, 5) and (2, 4). The endpoints are not included because the inequality is . The graph of the solution set is 876 54 32 1 0 1 2 3 4 5 6 7 8 The solution set of (c 2)(c 5)(c 4) 0 is (q, 5) ´ (2, 4). YOU TRY 4 Solve (y 3)(y 1)(y 1) 0. Graph the solution set, and write the solution in interval notation. 5 Solve a Rational Inequality An inequality containing a rational expression, p q , where p and q are polynomials, is called a rational inequality. The way we solve rational inequalities is very similar to the way we solve quadratic inequalities. www.mhhe.com/messersmith SECTION 10.7 Quadratic and Rational Inequalities 685
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