Ward Cummings
04/16/2024 · Senior High School

ESIGUALDADES CUADRATICAS \( \begin{array}{ll}\text { i. Resuelva las siguientes desigualdades } \\ \text { a) } x^{2}-6 x+8 \leq 0 & \text { b) }-2 x^{2}+1>0\end{array} \)

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Solve the equation \( x^{2}-6x+8 \leq 0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(x^{2}-6x+8\leq 0\) - step1: Rewrite the expression: \(x^{2}-6x+8=0\) - step2: Factor the expression: \(\left(x-4\right)\left(x-2\right)=0\) - step3: Separate into possible cases: \(\begin{align}&x-4=0\\&x-2=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=4\\&x=2\end{align}\) - step5: Determine the test intervals: \(\begin{align}&x<2\\&24\end{align}\) - step6: Choose a value: \(\begin{align}&x_{1}=1\\&x_{2}=3\\&x_{3}=5\end{align}\) - step7: Test the chosen value: \(\begin{align}&x<2\textrm{ }\textrm{is not a solution}\\&24\textrm{ }\textrm{is not a solution}\end{align}\) - step8: Include the critical value: \(\begin{align}&2\leq x\leq 4\textrm{ }\textrm{is the solution}\end{align}\) - step9: The final solution is \(2\leq x\leq 4:\) \(2\leq x\leq 4\) Solve the equation \( -2x^{2}+1>0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(-2x^{2}+1>0\) - step1: Rewrite the expression: \(-2x^{2}+1=0\) - step2: Move the constant to the right side: \(-2x^{2}=0-1\) - step3: Remove 0: \(-2x^{2}=-1\) - step4: Change the signs: \(2x^{2}=1\) - step5: Divide both sides: \(\frac{2x^{2}}{2}=\frac{1}{2}\) - step6: Divide the numbers: \(x^{2}=\frac{1}{2}\) - step7: Simplify the expression: \(x=\pm \sqrt{\frac{1}{2}}\) - step8: Simplify: \(x=\pm \frac{\sqrt{2}}{2}\) - step9: Separate into possible cases: \(\begin{align}&x=\frac{\sqrt{2}}{2}\\&x=-\frac{\sqrt{2}}{2}\end{align}\) - step10: Determine the test intervals: \(\begin{align}&x<-\frac{\sqrt{2}}{2}\\&-\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}\end{align}\) - step11: Choose a value: \(\begin{align}&x_{1}=-2\\&x_{2}=0\\&x_{3}=2\end{align}\) - step12: Test the chosen value: \(\begin{align}&x<-\frac{\sqrt{2}}{2}\textrm{ }\textrm{is not a solution}\\&-\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}\textrm{ }\textrm{is not a solution}\end{align}\) - step13: The final solution is \(-\frac{\sqrt{2}}{2}0\) es \(-\frac{\sqrt{2}}{2} < x < \frac{\sqrt{2}}{2}\).

Quick Answer

a) \(2 \leq x \leq 4\) b) \(-\frac{\sqrt{2}}{2} < x < \frac{\sqrt{2}}{2}\)
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