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\frac{x-2}{x+3}>-2
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- Eliminate Fraction by Multiplication: First, we need to get rid of the fraction by multiplying both sides of the inequality by , which is the denominator. However, we must consider that the inequality sign will change direction if we multiply by a negative number. Since we don't know if is positive or negative, we will have to consider two cases: one where is positive and one where is negative.
- Case : x > -3: Case : Assume (x+3) > 0, which means x > -3. Now multiply both sides of the inequality by to eliminate the fraction.\frac{x-2}{x+3} \cdot (x+3) > -2 \cdot (x+3)x - 2 > -2x - 6
- Combine Terms and Solve: Now, we will combine like terms and solve for .x + 2x > -6 + 23x > -4x > -\frac{4}{3}
- Case : x < -3: Case : Assume (x+3) < 0, which means x < -3. When we multiply both sides by a negative number, we must flip the inequality sign.\frac{x-2}{x+3} \cdot (x+3) < -2 \cdot (x+3)x - 2 < -2x - 6
- Combine Terms and Solve: Combine like terms and solve for in this case.x + 2x < -6 + 23x < -4x < -\frac{4}{3}However, this result contradicts our assumption that x < -3. Since is greater than , we cannot include this in our solution set.
- Combine Results from Both Cases: Now we need to combine the results from both cases. From Case , we have x > -\frac{4}{3}, and this is valid when x > -3. Therefore, the solution set for the inequality is x > -\frac{4}{3}, but we must exclude because the original inequality is undefined at .
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