Chapter 18: Chapter 18

Problem 539

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Find the solution set of \((\mathrm{x}+1) /(\mathrm{x}-2)>0\).

Short Answer

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The solution set for the inequality \(\frac{x+1}{x-2} > 0\) is \((-\infty, -1) \cup (2, \infty)\).

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TABLE OF CONTENTS

Step 1: Find critical points

To find critical points, we must determine where the inequality is equal to 0 and where it is undefined. In our inequality \(\frac{x+1}{x-2} > 0\), the numerator becomes 0 when \((-1)\), and the denominator is 0 at x = 2; at this point, the expression is undefined. Therefore, our critical points are \(x = -1\) and \(x = 2\).

Step 2: Identify intervals

Based on the critical points, we can divide the real number line into three intervals: 1. \(x < -1\) 2. \(-1 < x < 2\) 3. \(x > 2\)

Step 3: Test intervals in the inequality

Now we test each interval with a representative value to determine whether it satisfies the original inequality or not: Interval 1 (\(x < -1\)): Choose \(x = -2\). \(\frac{-2+1}{-2-2}>0⇔\frac{-1}{-4}>0\) True. Interval 2 (\(-1 < x < 2\)): Choose \(x = 0\). \(\frac{0+1}{0-2}>0⇔\frac{1}{-2}>0\) False Interval 3 (\(x > 2\)): Choose \(x = 3\). \(\frac{3+1}{3-2}>0⇔\frac{4}{1}>0\) True.

Step 4: Combine the results in the solution set

We combine the results from the intervals which satisfy the inequality: 1. Interval 1 (\(x < -1\)) 3. Interval 3 (\(x > 2\)) The solution set for the inequality \(\frac{x+1}{x-2} > 0\) is \((-\infty, -1) \cup (2, \infty)\).

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Chapter 18: Chapter 18

Find the solution set of \((\mathrm{x}+1) /(\mathrm{x}-2)>0\).

Short Answer

Step by step solution

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Step 1: Find critical points

Step 2: Identify intervals

Step 3: Test intervals in the inequality

Step 4: Combine the results in the solution set

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