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Problem 539

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

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