Frazier Franklin
08/30/2023 · Junior High School
Solve the inequality \( (x-2)(x-5)^{2}<0 \) Select all the intervals below that are included in the solution. \( \bigcirc_{x<5} \) \( \mathrm{x}_{\mathrm{x}}>5 \) \( \bigcirc_{x}<2 \) \( 2<\mathrm{x}<5 \)
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Step-by-step Solution
Solve the equation \( (x-2)(x-5)^{2}<0 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(\left(x-2\right)\left(x-5\right)^{2}<0\)
- step1: Rewrite the expression:
\(\left(x-2\right)\left(x-5\right)^{2}=0\)
- step2: Separate into possible cases:
\(\begin{align}&x-2=0\\&\left(x-5\right)^{2}=0\end{align}\)
- step3: Solve the equation:
\(\begin{align}&x=2\\&x=5\end{align}\)
- step4: Determine the test intervals:
\(\begin{align}&x<2\\&25\end{align}\)
- step5: Choose a value:
\(\begin{align}&x_{1}=1\\&x_{2}=4\\&x_{3}=6\end{align}\)
- step6: Test the chosen value:
\(\begin{align}&x<2\textrm{ }\textrm{is the solution}\\&25\textrm{ }\textrm{is not a solution}\end{align}\)
- step7: The final solution is \(x<2:\)
\(x<2\)
The solution to the inequality \( (x-2)(x-5)^{2}<0 \) is \( x<2 \).
Therefore, the interval that is included in the solution is \( \bigcirc_{x<2} \).
Quick Answer
The interval included in the solution is \( \bigcirc_{x<2} \).
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