Explanation Given: The inequality is x 3 − 4 x ≥ 0 . Properties of Inequality: For all real numbers a , b , and c : 1. If a < b , then a + c < b + c . 2. If a < b and if c > 0 , then a c < b c . 3. If a < b and if c < 0 , then a c < b c . Express the given inequality as, x 3 − 4 x ≥ 0 . Consider x 3 − 4 x = 0 . Factorise the cubic equation, x ( x 2 − 4 ) = 0 . x ( x 2 − 4 ) = x ( x 2 − 4 ) = x ( ( x ) 2 − ( 2 ) 2 ) = x ( ( x + 2 ) ( x − 2 ) ) Then, x = 0 , ( x + 2 ) = 0 or ( x − 2 ) = 0 . That is, x = 0 , x = − 2 or x = 2 . These three solutions determine four intervals on the number line: ( − ∞ , − 2 ) , ( − 2 , 0 ) , ( 0 , 2 ) and ( 2 , ∞ ) . Substitute a number from each interval into the original inequality to determine that the solution consists of the numbers that are greater than or equal to 0. For the interval ( − ∞ , − 2 ) , choose −3 to test for the variable x . x ( ( x + 2 ) ( x − 2 ) ) = ( − 3 ) ( ( ( − 3 ) + 2 ) ( ( − 3 ) − 2 ) ) = ( − 3 ) ( ( − 1 ) ( − 5 ) ) = ( − 3 ) ( 5 ) = − 15 This is not greater than or equal to 0, which is not considered as the solution. For the interval ( − 2 , 0 ) , choose −1 to test for the variable x

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Chapter R.5, Problem 39E

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Finite Mathematics and Calculus with Applications

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The inequality x 3 − 4 x ≥ 0 and graph the solution.

Solution Summary: The author explains how to solve the inequality x3-4x 0 and graph the solution.

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