Solve the Inequality for p |(2p-8)/4|>=9

$| \frac{2 p - 8}{4} | \geq 9$

Step 1

Write $| \frac{2 p - 8}{4} | \geq 9$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$\frac{2 p - 8}{4} \geq 0$

Solve the inequality.

Cancel the common factor of $2 p - 8$ and $4$ .

Factor $2$ out of $2 p$ .

$\frac{2 (p) - 8}{4} \geq 0$

Factor $2$ out of $- 8$ .

$\frac{2 p + 2 \cdot - 4}{4} \geq 0$

Factor $2$ out of $2 p + 2 \cdot - 4$ .

$\frac{2 (p - 4)}{4} \geq 0$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 (p - 4)}{2 (2)} \geq 0$

Cancel the common factor.

$\frac{2 (p - 4)}{2 \cdot 2} \geq 0$

Rewrite the expression.

$\frac{p - 4}{2} \geq 0$

Multiply both sides by $2$ .

$\frac{p - 4}{2} \cdot 2 \geq 0 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{p - 4}{2} \cdot 2 \geq 0 \cdot 2$

Rewrite the expression.

$p - 4 \geq 0 \cdot 2$

Simplify the right side.

Multiply $0$ by $2$ .

$p - 4 \geq 0$

Add $4$ to both sides of the inequality.

$p \geq 4$

In the piece where $\frac{2 p - 8}{4}$ is non-negative, remove the absolute value.

$\frac{2 p - 8}{4} \geq 9$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$\frac{2 p - 8}{4} < 0$

Solve the inequality.

Cancel the common factor of $2 p - 8$ and $4$ .

Factor $2$ out of $2 p$ .

$\frac{2 (p) - 8}{4} < 0$

Factor $2$ out of $- 8$ .

$\frac{2 p + 2 \cdot - 4}{4} < 0$

Factor $2$ out of $2 p + 2 \cdot - 4$ .

$\frac{2 (p - 4)}{4} < 0$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 (p - 4)}{2 (2)} < 0$

Cancel the common factor.

$\frac{2 (p - 4)}{2 \cdot 2} < 0$

Rewrite the expression.

$\frac{p - 4}{2} < 0$

Multiply both sides by $2$ .

$\frac{p - 4}{2} \cdot 2 < 0 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{p - 4}{2} \cdot 2 < 0 \cdot 2$

Rewrite the expression.

$p - 4 < 0 \cdot 2$

Simplify the right side.

Multiply $0$ by $2$ .

$p - 4 < 0$

Add $4$ to both sides of the inequality.

$p < 4$

In the piece where $\frac{2 p - 8}{4}$ is negative, remove the absolute value and multiply by $- 1$ .

$- \frac{2 p - 8}{4} \geq 9$

Write as a piecewise.

${\begin{cases} \frac{2 p - 8}{4} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Cancel the common factor of $2 p - 8$ and $4$ .

Factor $2$ out of $2 p$ .

${\begin{cases} \frac{2 (p) - 8}{4} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Factor $2$ out of $- 8$ .

${\begin{cases} \frac{2 p + 2 \cdot - 4}{4} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Factor $2$ out of $2 p + 2 \cdot - 4$ .

${\begin{cases} \frac{2 (p - 4)}{4} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Cancel the common factors.

Factor $2$ out of $4$ .

${\begin{cases} \frac{2 (p - 4)}{2 (2)} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Cancel the common factor.

${\begin{cases} \frac{2 (p - 4)}{2 \cdot 2} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Rewrite the expression.

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 p - 8}{4} \geq 9 & p < 4 \end{cases}$

Cancel the common factor of $2 p - 8$ and $4$ .

Factor $2$ out of $2 p$ .

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 (p) - 8}{4} \geq 9 & p < 4 \end{cases}$

Factor $2$ out of $- 8$ .

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 p + 2 \cdot - 4}{4} \geq 9 & p < 4 \end{cases}$

Factor $2$ out of $2 p + 2 \cdot - 4$ .

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 (p - 4)}{4} \geq 9 & p < 4 \end{cases}$

Cancel the common factors.

Factor $2$ out of $4$ .

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 (p - 4)}{2 (2)} \geq 9 & p < 4 \end{cases}$

Cancel the common factor.

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{2 (p - 4)}{2 \cdot 2} \geq 9 & p < 4 \end{cases}$

Rewrite the expression.

${\begin{cases} \frac{p - 4}{2} \geq 9 & p \geq 4 \\ - \frac{p - 4}{2} \geq 9 & p < 4 \end{cases}$

Step 2

Solve $\frac{p - 4}{2} \geq 9$ for $p$ .

Multiply both sides by $2$ .

$\frac{p - 4}{2} \cdot 2 \geq 9 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{p - 4}{2} \cdot 2 \geq 9 \cdot 2$

Rewrite the expression.

$p - 4 \geq 9 \cdot 2$

Simplify the right side.

Multiply $9$ by $2$ .

$p - 4 \geq 18$

Move all terms not containing $p$ to the right side of the inequality.

Add $4$ to both sides of the inequality.

$p \geq 18 + 4$

Add $18$ and $4$ .

$p \geq 22$

Step 3

Solve $- \frac{p - 4}{2} \geq 9$ for $p$ .

Divide each term in $- \frac{p - 4}{2} \geq 9$ by $- 1$ and simplify.

Divide each term in $- \frac{p - 4}{2} \geq 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{p - 4}{2}}{- 1} \leq \frac{9}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{p - 4}{2}}{1} \leq \frac{9}{- 1}$

Divide $\frac{p - 4}{2}$ by $1$ .

$\frac{p - 4}{2} \leq \frac{9}{- 1}$

Simplify the right side.

Divide $9$ by $- 1$ .

$\frac{p - 4}{2} \leq - 9$

Multiply both sides by $2$ .

$\frac{p - 4}{2} \cdot 2 \leq - 9 \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{p - 4}{2} \cdot 2 \leq - 9 \cdot 2$

Rewrite the expression.

$p - 4 \leq - 9 \cdot 2$

Simplify the right side.

Multiply $- 9$ by $2$ .

$p - 4 \leq - 18$

Move all terms not containing $p$ to the right side of the inequality.

Add $4$ to both sides of the inequality.

$p \leq - 18 + 4$

Add $- 18$ and $4$ .

$p \leq - 14$

Step 4

Find the union of the solutions.

$p \leq - 14$ or $p \geq 22$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$p \leq - 14 or p \geq 22$

Interval Notation:

$(- \infty, - 14] \cup [22, \infty)$

Step 6

Solve the Inequality for p |(2p-8)/4|>=9

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