Daniels Knight
09/07/2023 · Junior High School
Solve the inequality \( \frac{1}{x+9}>3 \) Give your answer in interval notation.
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Step-by-step Solution
Solve the inequality \( \frac{1}{x+9}>3 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(\frac{1}{x+9}>3\)
- step1: Find the domain:
\(\frac{1}{x+9}>3,x\neq -9\)
- step2: Move the expression to the left side:
\(\frac{1}{x+9}-3>0\)
- step3: Subtract the terms:
\(-\frac{26+3x}{x+9}>0\)
- step4: Change the sign:
\(\frac{26+3x}{x+9}<0\)
- step5: Set the numerator and denominator of \(\frac{26+3x}{x+9}\) equal to 0\(:\)
\(\begin{align}&26+3x=0\\&x+9=0\end{align}\)
- step6: Calculate:
\(\begin{align}&x=-\frac{26}{3}\\&x=-9\end{align}\)
- step7: Determine the test intervals:
\(\begin{align}&x<-9\\&-9-\frac{26}{3}\end{align}\)
- step8: Choose a value:
\(\begin{align}&x_{1}=-10\\&x_{2}=-\frac{53}{6}\\&x_{3}=-8\end{align}\)
- step9: Test the chosen value:
\(\begin{align}&x<-9\textrm{ }\textrm{is not a solution}\\&-9-\frac{26}{3}\textrm{ }\textrm{is not a solution}\end{align}\)
- step10: The final solution is \(-93 \) is \( -9 < x < -\frac{26}{3} \) in interval notation.
Quick Answer
The solution is \( -9 < x < -\frac{26}{3} \).
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