Explanation Given information: The inequality is 2 x 3 − 18 x < x 2 − 9 . Calculation: The given inequality is 2 x 3 − 18 x < x 2 − 9 . Rearrange the above inequality as, 2 x 3 − x 2 − 18 x + 9 < 0 Consider P ( x ) = 2 x 3 − x 2 − 18 x + 9 , then factorize the polynomial P ( x ) as, P ( x ) = 2 x 3 − x 2 − 18 x + 9 = x 2 ( 2 x − 1 ) − 9 ( 2 x − 1 ) = ( 2 x − 1 ) ( x 2 − 9 ) Further calculate the above expression, P ( x ) = ( 2 x − 1 ) { x 2 − ( 3 ) 2 } = ( 2 x − 1 ) ( x + 3 ) ( x − 3 ) Therefore, the factorize form of the polynomial is P ( x ) = ( 2 x − 1 ) ( x + 3 ) ( x − 3 ) . Then the inequality becomes P ( x ) < 0 . The zeros of the polynomial P ( x ) is calculated as, P ( x ) = 0 ( 2 x − 1 ) ( x + 3 ) ( x − 3 ) = 0 2 x − 1 = 0 or x + 3 = 0 or x − 3 = 0 x = 1 2 or x = − 3 or x = 3 Therefore, the real zeros of the polynomial P ( x ) are − 3 , 1 2 and 3 . Hence, the intervals determined by the zeros of the polynomial are ( − ∞ , − 3 ) , ( − 3 , 1 2 ) , ( 1 2 , 3 ) and ( 3 , ∞ ) . The sign of each factor, in each interval are shown in Table 1. Sign of ( − ∞ , − 3 ) ( − 3 , 1 2 ) ( 1 2 , 3 ) ( 3 , ∞ ) ( 2 x − 1 ) − − + + ( x + 3 ) − + + + x − 3 − − − + P ( x ) = ( 2 x − 1 ) ( x + 3 ) ( x − 3 ) − + − + Table 1 From the Table 1, it is shown that the inequality P ( x ) < 0 satisfies in the intervals ( − ∞ , − 3 ) and ( 1 2 , 3 ) . The end points of the intervals ( − ∞ , − 3 ) , ( − 3 , 1 2 ) , ( 1 2 , 3 ) and ( 3 , ∞ ) , are − 3 , 1 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 − 3 , substitute − 3 for x in the inequality P ( x ) < 0 to check whether the point − 3 satisfies the inequality P ( x ) < 0 or not as, P ( − 3 ) < ? 0 { 2 ( − 3 ) − 1 } ( − 3 + 3 ) ( − 3 − 3 ) < ? 0 ( − 7 ) × 0 × ( − 6 ) < ? 0 0 < 0 Hence, the inequality P ( x ) < 0 does not satisfy at the point x = − 3

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 3.7, Problem 8E

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The inequality 2x3−18x<x2−9.

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