Griffin Parsons
06/08/2024 · Primary School
Solve the inequality \( \frac{9}{x+1} \geq-5 \). Use exact values when entering the answer. The solution set is: Get help: Video Video
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Step-by-step Solution
Solve the equation \( \frac{9}{x+1} \geq -5 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(\frac{9}{x+1}\geq -5\)
- step1: Find the domain:
\(\frac{9}{x+1}\geq -5,x\neq -1\)
- step2: Move the expression to the left side:
\(\frac{9}{x+1}-\left(-5\right)\geq 0\)
- step3: Subtract the terms:
\(\frac{14+5x}{x+1}\geq 0\)
- step4: Set the numerator and denominator of \(\frac{14+5x}{x+1}\) equal to 0\(:\)
\(\begin{align}&14+5x=0\\&x+1=0\end{align}\)
- step5: Calculate:
\(\begin{align}&x=-\frac{14}{5}\\&x=-1\end{align}\)
- step6: Determine the test intervals:
\(\begin{align}&x<-\frac{14}{5}\\&-\frac{14}{5}-1\end{align}\)
- step7: Choose a value:
\(\begin{align}&x_{1}=-4\\&x_{2}=-2\\&x_{3}=0\end{align}\)
- step8: Test the chosen value:
\(\begin{align}&x<-\frac{14}{5}\textrm{ }\textrm{is the solution}\\&-\frac{14}{5}-1\textrm{ }\textrm{is the solution}\end{align}\)
- step9: Include the critical value:
\(\begin{align}&x\leq -\frac{14}{5}\textrm{ }\textrm{is the solution}\\&x>-1\textrm{ }\textrm{is the solution}\end{align}\)
- step10: The final solution is \(x \in \left(-\infty,-\frac{14}{5}\right]\cup \left(-1,+\infty\right):\)
\(x \in \left(-\infty,-\frac{14}{5}\right]\cup \left(-1,+\infty\right)\)
- step11: Check if the solution is in the defined range:
\(x \in \left(-\infty,-\frac{14}{5}\right]\cup \left(-1,+\infty\right),x\neq -1\)
- step12: Find the intersection:
\(x \in \left(-\infty,-\frac{14}{5}\right]\cup \left(-1,+\infty\right)\)
The solution to the inequality \( \frac{9}{x+1} \geq -5 \) is \( x \in (-\infty,-\frac{14}{5}]\cup (-1,+\infty) \).
Quick Answer
\( x \in (-\infty,-\frac{14}{5}]\cup (-1,+\infty) \)
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