Allan Peterson
09/22/2023 · Middle School
Solve the inequality: \( \frac{x+2}{x-8} \geq 0 \) Interval notation solution: No solution
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \frac{x+2}{x-8} \geq 0 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(\frac{x+2}{x-8}\geq 0\)
- step1: Find the domain:
\(\frac{x+2}{x-8}\geq 0,x\neq 8\)
- step2: Set the numerator and denominator of \(\frac{x+2}{x-8}\) equal to 0\(:\)
\(\begin{align}&x+2=0\\&x-8=0\end{align}\)
- step3: Calculate:
\(\begin{align}&x=-2\\&x=8\end{align}\)
- step4: Determine the test intervals:
\(\begin{align}&x<-2\\&-28\end{align}\)
- step5: Choose a value:
\(\begin{align}&x_{1}=-3\\&x_{2}=3\\&x_{3}=9\end{align}\)
- step6: Test the chosen value:
\(\begin{align}&x<-2\textrm{ }\textrm{is the solution}\\&-28\textrm{ }\textrm{is the solution}\end{align}\)
- step7: Include the critical value:
\(\begin{align}&x\leq -2\textrm{ }\textrm{is the solution}\\&x>8\textrm{ }\textrm{is the solution}\end{align}\)
- step8: The final solution is \(x \in \left(-\infty,-2\right]\cup \left(8,+\infty\right):\)
\(x \in \left(-\infty,-2\right]\cup \left(8,+\infty\right)\)
- step9: Check if the solution is in the defined range:
\(x \in \left(-\infty,-2\right]\cup \left(8,+\infty\right),x\neq 8\)
- step10: Find the intersection:
\(x \in \left(-\infty,-2\right]\cup \left(8,+\infty\right)\)
The solution to the inequality \( \frac{x+2}{x-8} \geq 0 \) is \( x \in (-\infty,-2\right]\cup (8,+\infty) \).
Therefore, the interval notation solution is \( x \in (-\infty,-2\right]\cup (8,+\infty) \).
Quick Answer
Interval notation solution: \( x \in (-\infty,-2\right]\cup (8,+\infty) \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit