Explanation Given information: The inequality is ( x − 3 ) ( x + 5 ) ( 2 x + 5 ) < 0 . Calculation: The given inequality is ( x − 3 ) ( x + 5 ) ( 2 x + 5 ) < 0 . Consider ( x − 3 ) ( x + 5 ) ( 2 x + 5 ) as P ( x ) , then the inequality becomes P ( x ) < 0 . The zeros of the polynomial P ( x ) is calculated as, P ( x ) = 0 ( x − 3 ) ( x + 5 ) ( 2 x + 5 ) = 0 x − 3 = 0 or x + 5 = 0 or 2 x + 5 = 0 x = 3 or x = − 5 or 2 x = − 5 Further calculate the above inequality, x = 3 or x = − 5 or x = − 5 2 Therefore, the real zeros of the polynomial P ( x ) are − 5 , − 5 2 and 3 . Hence, the intervals determined by the zeros of the polynomial are ( − ∞ , − 5 ) , ( − 5 , − 5 2 ) , ( − 5 2 , 3 ) and ( 3 , ∞ ) . The sign of each factor, in each interval are shown in Table 1. Sign of ( − ∞ , − 5 ) ( − 5 , − 5 2 ) ( − 5 2 , 3 ) ( 3 , ∞ ) x − 3 − − − + x + 5 − + + + 2 x + 3 − − + + ( x − 3 ) ( x + 5 ) ( 2 x + 5 ) − + − + Table 1 From the Table 1, it is shown that the inequality P ( x ) < 0 satisfies in the intervals ( − ∞ , − 5 ) and ( − 5 2 , 3 ) . The end points of the intervals ( − ∞ , − 5 ) and ( − 5 2 , 3 ) are − 5 , − 5 2 and 3 . To check whether the end points satisfies the inequality P ( x ) < 0 or not, substitute the end points for x in the inequality P ( x ) < 0 respectively. For the point − 5 , substitute − 5 for x in the inequality P ( x ) < 0 to check whether the point − 5 satisfies the inequality P ( x ) < 0 or not as, P ( − 5 ) < ? 0 ( − 5 − 3 ) ( − 5 + 5 ) ( 2 ( − 5 ) + 5 ) < ? 0 ( − 8 ) × 0 × ( − 5 ) < ? 0 0 < 0 Hence, the inequality P ( x ) < 0 does not satisfy at the point x = − 5

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 3.7, Problem 3E

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The inequality (x−3)(x+5)(2x+5)<0.

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Chapter 3.7, Problem 2E

Chapter 3.7, Problem 4E

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