Fitzgerald Parsons
02/04/2023 · Middle School
Solve the polynomial inequality and graph \( 2 x^{2}-9 x \geq 0 \)
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Step-by-step Solution
Solve the equation \( 2x^{2}-9x\geq 0 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(2x^{2}-9x\geq 0\)
- step1: Rewrite the expression:
\(2x^{2}-9x=0\)
- step2: Factor the expression:
\(x\left(2x-9\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x=0\\&2x-9=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=0\\&x=\frac{9}{2}\end{align}\)
- step5: Determine the test intervals:
\(\begin{align}&x<0\\&0\frac{9}{2}\end{align}\)
- step6: Choose a value:
\(\begin{align}&x_{1}=-1\\&x_{2}=2\\&x_{3}=6\end{align}\)
- step7: Test the chosen value:
\(\begin{align}&x<0\textrm{ }\textrm{is the solution}\\&0\frac{9}{2}\textrm{ }\textrm{is the solution}\end{align}\)
- step8: Include the critical value:
\(\begin{align}&x\leq 0\textrm{ }\textrm{is the solution}\\&x\geq \frac{9}{2}\textrm{ }\textrm{is the solution}\end{align}\)
- step9: The final solution is \(x \in \left(-\infty,0\right]\cup \left[\frac{9}{2},+\infty\right):\)
\(x \in \left(-\infty,0\right]\cup \left[\frac{9}{2},+\infty\right)\)
The solution to the polynomial inequality \(2x^{2}-9x \geq 0\) is \(x \in (-\infty,0] \cup [\frac{9}{2},+\infty)\).
Unfortunately, I am unable to plot the graph for you at the moment.
Quick Answer
The solution is \(x \in (-\infty,0] \cup [\frac{9}{2},+\infty)\).
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