Nunez Sullivan
07/21/2023 · Senior High School

Solve the following inequality and graph the solution: \[ (x-8)(x+3)>0 \] Choose test values and indicate whether the inequality is true o false in each region.

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Step-by-step Solution

Solve the equation \( (x-8)(x+3)>0 \). Solve the inequality by following steps: - step0: Solve the inequality by testing the values in the interval: \(\left(x-8\right)\left(x+3\right)>0\) - step1: Rewrite the expression: \(\left(x-8\right)\left(x+3\right)=0\) - step2: Separate into possible cases: \(\begin{align}&x-8=0\\&x+3=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=8\\&x=-3\end{align}\) - step4: Determine the test intervals: \(\begin{align}&x<-3\\&-38\end{align}\) - step5: Choose a value: \(\begin{align}&x_{1}=-4\\&x_{2}=3\\&x_{3}=9\end{align}\) - step6: Test the chosen value: \(\begin{align}&x<-3\textrm{ }\textrm{is the solution}\\&-38\textrm{ }\textrm{is the solution}\end{align}\) - step7: The final solution is \(x \in \left(-\infty,-3\right)\cup \left(8,+\infty\right):\) \(x \in \left(-\infty,-3\right)\cup \left(8,+\infty\right)\) The solution to the inequality \((x-8)(x+3)>0\) is \(x \in (-\infty,-3) \cup (8,+\infty)\). To graph the solution, we can choose test values in each region and determine whether the inequality is true or false in each region. 1. For \(x < -3\), let's choose \(x = -4\): \((-4-8)(-4+3) = (-12)(-1) = 12 > 0\) The inequality is true for \(x < -3\). 2. For \(-3 < x < 8\), let's choose \(x = 0\): \((0-8)(0+3) = (-8)(3) = -24 < 0\) The inequality is false for \(-3 < x < 8\). 3. For \(x > 8\), let's choose \(x = 9\): \((9-8)(9+3) = (1)(12) = 12 > 0\) The inequality is true for \(x > 8\). Therefore, the solution to the inequality \((x-8)(x+3)>0\) is \(x \in (-\infty,-3) \cup (8,+\infty)\).

Quick Answer

The solution to the inequality \((x-8)(x+3)>0\) is \(x \in (-\infty,-3) \cup (8,+\infty)\). Test values show the inequality is true for \(x < -3\) and \(x > 8\), and false for \(-3 < x < 8\).
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