Explanation Given information: The rational inequality is x 2 ≥ 5 x + 1 + 4 . Procedure used: In order to solve for any rational inequality, the following steps are followed: Step 1: Move all the term to one side and rewrite the inequality so that all non-zero terms appear on one side of the inequality and bring all quotients to a common denominator. Step 2: Factor the numerator and denominator into irreducible factors, and then find the cut points. Step 3: Find and list the intervals determined by the cut-points. Step 4: Make a sign table and use test values to make table of the signs of each factor in each interval. In the last row of the table determine the sign of the rational function on that interval. Step 5: Solve for the solution of inequality from the last row of the table. Check whether the endpoints of these intervals satisfy the inequality. Calculation: The rational inequality is x 2 ≥ 5 x + 1 + 4 . Step 1: Move all non-zero terms to left hand side. x 2 − 5 x + 1 − 4 ≥ 0 Make common denominator and simplify as, x ( x + 1 ) − 10 − 8 ( x + 1 ) 2 ( x + 1 ) ≥ 0 x 2 + x − 10 − 8 x − 8 2 ( x + 1 ) ≥ 0 x 2 − 7 x − 18 2 ( x + 1 ) ≥ 0 The left-hand side of inequality is a rational function. Step 2: Factor the numerator and denominator as, x 2 − 7 x − 18 2 ( x + 1 ) ≥ 0 x 2 − 9 x + 2 x − 18 2 ( x + 1 ) ≥ 0 x ( x − 9 ) + 2 ( x − 9 ) 2 ( x + 1 ) ≥ 0 ( x − 9 ) ( x + 2 ) 2 ( x + 1 ) ≥ 0 So, the zeros of numerator is 9 and − 2 and zeros of denominator are − 1 . Therefore the cut-points are x = − 2 , x = − 1 , and x = 9 . Step 3: The interval determined by cut-points are ( − ∞ , − 2 ) , ( − 2 , − 1 ) , ( − 1 , 9 ) , ( 9 , ∞ )

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 3.7, Problem 34E

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The rational inequality x2≥5x+1+4.

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Chapter 3.7, Problem 33E

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