Pritchard Fleming
10/03/2023 · Elementary School
Solve the polynomial inequality and graph the solution set on a real number line. Express the solu \( 5 x^{2}+18 x-8 \leq 0 \) Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill The solution set is A. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using ra the expression.)
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Step-by-step Solution
Solve the equation \( 5x^{2}+18x-8 \leq 0 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(5x^{2}+18x-8\leq 0\)
- step1: Rewrite the expression:
\(5x^{2}+18x-8=0\)
- step2: Factor the expression:
\(\left(x+4\right)\left(5x-2\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x+4=0\\&5x-2=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=-4\\&x=\frac{2}{5}\end{align}\)
- step5: Determine the test intervals:
\(\begin{align}&x<-4\\&-4\frac{2}{5}\end{align}\)
- step6: Choose a value:
\(\begin{align}&x_{1}=-5\\&x_{2}=-2\\&x_{3}=1\end{align}\)
- step7: Test the chosen value:
\(\begin{align}&x<-4\textrm{ }\textrm{is not a solution}\\&-4\frac{2}{5}\textrm{ }\textrm{is not a solution}\end{align}\)
- step8: Include the critical value:
\(\begin{align}&-4\leq x\leq \frac{2}{5}\textrm{ }\textrm{is the solution}\end{align}\)
- step9: The final solution is \(-4\leq x\leq \frac{2}{5}:\)
\(-4\leq x\leq \frac{2}{5}\)
The solution to the polynomial inequality \(5x^{2}+18x-8 \leq 0\) is \(-4 \leq x \leq \frac{2}{5}\).
Therefore, the solution set is \([-4, \frac{2}{5}]\).
Quick Answer
The solution set is \([-4, \frac{2}{5}]\).
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