Explanation Procedure used: (1) . Find an equivalent inequality with 0 on one side. (2). Change the inequality symbol to an equals sign and solve the related equation. (3). Find values of the variables for which the related rational function is not defined. (4). The numbers found in steps (2) and (3) are called the critical values. Use the critical values to divide the x - axis into intervals. Then determine the function’s sign in each interval using an x value from the interval or using the graph of the equation. (5) Select the intervals for which the inequality is satisfied and write intervals notation or set builder notation for the solution set. If the inequality symbol ≤ or ≥ , then the solutions to step (2) should be include in the solution set. The x values found in step (3) are never included in the solution set. Calculation: The given polynomial inequality is h ( x ) ≥ 0 , where h ( x ) = 7 x ( x − 1 ) ( x + 5 ) . By the above procedure, firstly obtain the solution set of the inequality. That is, obtain the solution of the equation 7 x ( x − 1 ) ( x + 5 ) = 0 . The rational equation is undefined at the points x = − 5 and x = 1 . Obtain the critical point of the equation 7 x ( x − 1 ) ( x + 5 ) = 0 by multiplying ( x − 1 ) ( x + 5 ) on both sides. 7 x ( x − 1 ) ( x + 5 ) = 0 ( x − 1 ) ( x + 5 ) ⋅ ( 7 x ( x − 1 ) ( x + 5 ) ) = 0 ⋅ ( x − 1 ) ( x + 5 ) 7 x = 0 x = 0 Thus, the critical values of the rational equation are 0 , − 5 , and 1 . The critical values divide the x axis into the intervals as ( − ∞ , − 5 ) , ( − 5 , 0 ) , ( 0 , 1 ) , and ( 1 , ∞ ) . Find the sign of the rational function on each interval by selecting test value as follows. Obtain the sign of polynomial h ( x ) = 7 x ( x − 1 ) ( x + 5 ) at x = − 6 on the interval ( − ∞ , − 5 ) . h ( − 6 ) = 7 ( − 6 ) ( − 6 − 1 ) ( − 6 + 5 ) = − 42 ( − 7 ) ( − 1 ) = − 6 < 0 Obtain the sign of polynomial h ( x ) = 7 x ( x − 1 ) ( x + 5 ) at x = 1 on the interval ( − 5 , 0 )

College Algebra: Graphs and Models (6th Edition)

6th Edition

ISBN: 9780134179032

Author: Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen, Judith A. Penna

Publisher: PEARSON

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Chapter 4.6, Problem 13E

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To solve: The rational inequality h(x)≥0, where h(x)=7x(x−1)(x+5).

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Chapter 4.6, Problem 12E

Chapter 4.6, Problem 14E

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College Algebra: Graphs and Models (6th Edition)

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The rational inequality h ( x ) ≥ 0 , where h ( x ) = 7 x ( x − 1 ) ( x + 5 ) .

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To solve: The rational inequality h(x)≥0, where h(x)=7x(x−1)(x+5).

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