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 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 include in the solution set. Calculation: The given polynomial inequality is x + 1 x − 2 ≥ 3 . By the procedure from 1 to 5, find the solution set of the inequality. An equivalent inequality with 0 on one side is x + 1 x − 2 − 3 ≥ 0 Change the inequlity sign by equal sign to form related equation as x + 1 x − 2 − 3 = 0 . The rational equation is undefined at x = 2 . Solve the above equation for critical value by multiplying ( x − 2 ) on both sides. ( x − 2 ) ⋅ ( x + 1 x − 2 − 3 ) = 0 ⋅ ( x − 2 ) x + 1 − 3 ( x − 2 ) = 0 − 2 x + 7 = 0 x = 7 2 Thus, the critical values of the rational equation are 7 2 and 2 . The critical values divide the x axis into the intervals as ( − ∞ , 2 ) , ( 2 , 7 2 ) and ( 7 2 , ∞ ) . Find the sign of the rational function on each interval by selecting test value as follows. Let the rational function be denoted by g ( x ) = x + 1 x − 2 − 3 . Obtain the sign of polynomial g ( x ) = x + 1 x − 2 − 3 at x = 0 on the interval ( − ∞ , 2 )

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 59E

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To solve: The rational inequality x+1x−2≥3 by listing the critical values of the related function.

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Chapter 4.6, Problem 58E

Chapter 4.6, Problem 60E

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College Algebra: Graphs and Models (6th Edition)

Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

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The rational inequality x + 1 x − 2 ≥ 3 by listing the critical values of the related function.

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