SECTION 4.7 Polynomial and Rational Inequalities 263 SUMMARY Steps for Solving Polynomial and Rational Inequalities Algebraically Step 1 Write the inequality so that a polynomial or rational function f is on the left side and zero is on the right side in one of the following forms: ( ) ( ) ( ) ( ) > ≥ < ≤ f x f x f x f x 0 0 0 0 For rational functions, be sure that the left side is written as a single quotient. Find the domain of f . Step 2 Determine the real numbers at which ( ) = f x 0 and, if the function is rational, the real numbers at which the function f is undefined. Step 3 Use the numbers found in Step 2 to divide the real number line into intervals. Step 4 Select a number in each interval and evaluate f at the number. • If the value of f is positive, then ( ) > f x 0 for all numbers x in the interval. • If the value of f is negative, then ( ) < f x 0 for all numbers x in the interval. • If the inequality is not strict ( ) ≥ ≤ or , include the solutions of ( ) = f x 0 that are in the domain of f in the solution set. Be careful to exclude values of x where f is undefined. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 4.7 Assess Your Understanding 1. Solve the inequality − > x 3 4 5. Graph the solution set. (pp. A80–A82) 2. Solve the inequality − ≤ x x5 24 2 . Graph the solution set. (pp. 180–181) 3. Multiple Choice Which of the following could be a test number for the interval − < < x 2 5? (a) −3 (b) −2 (c) 4 (d) 7 4. True or False The graph of ( ) = − f x x x 3 is above the x- axis for < x 0 or > x 3, so the solution set of the inequality − ≥ x x 3 0 is { } ≤ ≥ x x x 0 or 3 . Concepts and Vocabulary Skill Building In Problems 5–8, use the graph of the function f to solve the inequality. 5. (a) ( ) > f x 0 (b) ( ) ≤ f x 0 x y 0 1 2 3 2 1 –1 –2 –2 –1 6. (a) ( ) < f x 0 (b) ( ) ≥ f x 0 x y 2 3 2 1 –1 –2 1 –2 –1 7. (a) ( ) < f x 0 (b) ( ) ≥ f x 0 x y 23 3 3 23 x 5 21 x 5 1 y 5 0 8. (a) ( ) > f x 0 (b) ( ) ≤ f x 0 x 5 3 1 24 22 4 23 y 3 2 21 x 5 21 x 5 2 y 5 1 22 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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