Allan Collins
05/13/2024 · High School
Решите неравенство: \( 3 x^{2}-14 x-5 \leq 0 \) Выберите ответ: \( x \in\left[-\frac{1}{3} ; 5\right] \) \( x \in\left(-\infty ;-\frac{1}{3}\right] \cup[5 ; \infty) \) \( x \in\left(-\infty ;-\frac{1}{3}\right) \cup(5 ; \infty) \) \( x \in \emptyset \)
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Solve the equation \( 3x^{2}-14x-5 \leq 0 \).
Solve the inequality by following steps:
- step0: Solve the inequality by testing the values in the interval:
\(3x^{2}-14x-5\leq 0\)
- step1: Rewrite the expression:
\(3x^{2}-14x-5=0\)
- step2: Factor the expression:
\(\left(x-5\right)\left(3x+1\right)=0\)
- step3: Separate into possible cases:
\(\begin{align}&x-5=0\\&3x+1=0\end{align}\)
- step4: Solve the equation:
\(\begin{align}&x=5\\&x=-\frac{1}{3}\end{align}\)
- step5: Determine the test intervals:
\(\begin{align}&x<-\frac{1}{3}\\&-\frac{1}{3}5\end{align}\)
- step6: Choose a value:
\(\begin{align}&x_{1}=-2\\&x_{2}=3\\&x_{3}=6\end{align}\)
- step7: Test the chosen value:
\(\begin{align}&x<-\frac{1}{3}\textrm{ }\textrm{is not a solution}\\&-\frac{1}{3}5\textrm{ }\textrm{is not a solution}\end{align}\)
- step8: Include the critical value:
\(\begin{align}&-\frac{1}{3}\leq x\leq 5\textrm{ }\textrm{is the solution}\end{align}\)
- step9: The final solution is \(-\frac{1}{3}\leq x\leq 5:\)
\(-\frac{1}{3}\leq x\leq 5\)
Решение неравенства \(3x^{2}-14x-5 \leq 0\) равно \(x \in \left[-\frac{1}{3}; 5\right]\).
Следовательно, правильный ответ: \(x \in \left[-\frac{1}{3}; 5\right]\).
Quick Answer
Решение неравенства: \(x \in \left[-\frac{1}{3}; 5\right]\).
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