Questions: Solve the inequality. Write the solution set in interval notation if possible. [ fracp^2+p-12p+7 geq 0 ] The solution set is (, ) (, ] [, ) [, ] -infty infty cup 5

$Solve the inequality. Write the solution set in interval notation if possible. [ fracp^2+p-12p+7 geq 0 ] The solution set is (, ) (, ] [, ) [, ] -infty infty cup 5$

Transcript text: Solve the inequality. Write the solution set in interval notation if possible. \[ \frac{p^{2}+p-12}{p+7} \geq 0 \] The solution set is $\square$ \begin{tabular}{|ccc|} \hline$(\square, \square)$ & $(\square, \square]$ & {$[\square, \square)$} \\ {$[\square, \square]$} & $-\infty$ & $\infty$ \\ $\square$ & $\square \cup \square$ & $\}$ \\ \hline$\square$ & 5 \\ \hline \end{tabular}

Solution

Solution Steps

To solve the inequality $\frac{p^{2}+p-12}{p+7} \geq 0$, we need to find the values of $p$ that make the expression non-negative. First, factor the numerator $p^2 + p - 12$ to find its roots. Then, identify the critical points from the roots of the numerator and the point where the denominator is zero. Use these critical points to divide the number line into intervals. Test each interval to determine where the inequality holds true. Finally, express the solution set in interval notation.

Step 1: Factor the Numerator

The inequality we need to solve is

\[ \frac{p^{2}+p-12}{p+7} \geq 0. \]

First, we factor the numerator $p^2 + p - 12$. The factors are $(p - 3)(p + 4)$. Thus, we can rewrite the inequality as

\[ \frac{(p - 3)(p + 4)}{p + 7} \geq 0. \]

Step 2: Identify Critical Points

Next, we find the critical points by setting the numerator and denominator to zero:

$p - 3 = 0 \Rightarrow p = 3$
$p + 4 = 0 \Rightarrow p = -4$
$p + 7 = 0 \Rightarrow p = -7$

The critical points are $p = -7$, $p = -4$, and $p = 3$.

Step 3: Test Intervals

We divide the number line into intervals based on the critical points:

$(- \infty, -7)$
$(-7, -4)$
$(-4, 3)$
$(3, \infty)$

We test each interval to determine where the inequality holds true.

For $p < -7$, choose $p = -8$: \[ \frac{(-8 - 3)(-8 + 4)}{-8 + 7} = \frac{(-11)(-4)}{-1} > 0. \]
For $-7 < p < -4$, choose $p = -5$: \[ \frac{(-5 - 3)(-5 + 4)}{-5 + 7} = \frac{(-8)(-1)}{2} > 0. \]
For $-4 < p < 3$, choose $p = 0$: \[ \frac{(0 - 3)(0 + 4)}{0 + 7} = \frac{(-3)(4)}{7} < 0. \]
For $p > 3$, choose $p = 4$: \[ \frac{(4 - 3)(4 + 4)}{4 + 7} = \frac{(1)(8)}{11} > 0. \]

Step 4: Combine Results

The inequality holds true in the intervals $(- \infty, -7)$, $(-7, -4)$, and $(3, \infty)$. We also include the points where the numerator is zero, which are $p = -4$ and $p = 3$. However, $p = -7$ is excluded since it makes the denominator zero.

Thus, the solution set in interval notation is

\[ (-\infty, -7) \cup (-7, -4] \cup [3, \infty). \]