Marsh Warner
11/22/2023 · High School

Solve the inequality \( \frac{x+2}{x+9}>3 \) Give your answer in interval notation. Pet help: Video

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Solve the inequality \( \frac{x+2}{x+9}>3 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(\frac{x+2}{x+9}>3\) - step1: Find the domain: \(\frac{x+2}{x+9}>3,x\neq -9\) - step2: Move the expression to the left side: \(\frac{x+2}{x+9}-3>0\) - step3: Subtract the terms: \(-\frac{2x+25}{x+9}>0\) - step4: Change the sign: \(\frac{2x+25}{x+9}<0\) - step5: Set the numerator and denominator of \(\frac{2x+25}{x+9}\) equal to 0\(:\) \(\begin{align}&2x+25=0\\&x+9=0\end{align}\) - step6: Calculate: \(\begin{align}&x=-\frac{25}{2}\\&x=-9\end{align}\) - step7: Determine the test intervals: \(\begin{align}&x<-\frac{25}{2}\\&-\frac{25}{2}-9\end{align}\) - step8: Choose a value: \(\begin{align}&x_{1}=-14\\&x_{2}=-11\\&x_{3}=-8\end{align}\) - step9: Test the chosen value: \(\begin{align}&x<-\frac{25}{2}\textrm{ }\textrm{is not a solution}\\&-\frac{25}{2}-9\textrm{ }\textrm{is not a solution}\end{align}\) - step10: The final solution is \(-\frac{25}{2}3 \) is \( x \in (-\frac{25}{2}, -9) \).

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\( x \in (-\frac{25}{2}, -9) \)
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