Solve the Inequality for p 2-9|p+7|>-16

$2 - 9 | p + 7 | > - 16$

Step 1

Write $2 - 9 | p + 7 | > - 16$ as a piecewise.

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

$p + 7 \geq 0$

Subtract $7$ from both sides of the inequality.

$p \geq - 7$

In the piece where $p + 7$ is non-negative, remove the absolute value.

$2 - 9 (p + 7) > - 16$

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

$p + 7 < 0$

Subtract $7$ from both sides of the inequality.

$p < - 7$

In the piece where $p + 7$ is negative, remove the absolute value and multiply by $- 1$ .

$2 - 9 (- (p + 7)) > - 16$

Write as a piecewise.

${\begin{cases} 2 - 9 (p + 7) > - 16 & p \geq - 7 \\ 2 - 9 (- (p + 7)) > - 16 & p < - 7 \end{cases}$

Simplify $2 - 9 (p + 7) > - 16$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 - 9 p - 9 \cdot 7 > - 16 & p \geq - 7 \\ 2 - 9 (- (p + 7)) > - 16 & p < - 7 \end{cases}$

Multiply $- 9$ by $7$ .

${\begin{cases} 2 - 9 p - 63 > - 16 & p \geq - 7 \\ 2 - 9 (- (p + 7)) > - 16 & p < - 7 \end{cases}$

Subtract $63$ from $2$ .

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 - 9 (- (p + 7)) > - 16 & p < - 7 \end{cases}$

Simplify $2 - 9 (- (p + 7)) > - 16$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 - 9 (- p - 1 \cdot 7) > - 16 & p < - 7 \end{cases}$

Multiply $- 1$ by $7$ .

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 - 9 (- p - 7) > - 16 & p < - 7 \end{cases}$

Apply the distributive property.

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 - 9 (- p) - 9 \cdot - 7 > - 16 & p < - 7 \end{cases}$

Multiply $- 1$ by $- 9$ .

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 + 9 p - 9 \cdot - 7 > - 16 & p < - 7 \end{cases}$

Multiply $- 9$ by $- 7$ .

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 2 + 9 p + 63 > - 16 & p < - 7 \end{cases}$

Add $2$ and $63$ .

${\begin{cases} - 9 p - 61 > - 16 & p \geq - 7 \\ 9 p + 65 > - 16 & p < - 7 \end{cases}$

Step 2

Solve $- 9 p - 61 > - 16$ when $p \geq - 7$ .

Solve $- 9 p - 61 > - 16$ for $p$ .

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

Add $61$ to both sides of the inequality.

$- 9 p > - 16 + 61$

Add $- 16$ and $61$ .

$- 9 p > 45$

Divide each term in $- 9 p > 45$ by $- 9$ and simplify.

Divide each term in $- 9 p > 45$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 p}{- 9} < \frac{45}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 p}{- 9} < \frac{45}{- 9}$

Divide $p$ by $1$ .

$p < \frac{45}{- 9}$

Simplify the right side.

Divide $45$ by $- 9$ .

$p < - 5$

Find the intersection of $p < - 5$ and $p \geq - 7$ .

$- 7 \leq p < - 5$

Step 3

Solve $9 p + 65 > - 16$ when $p < - 7$ .

Solve $9 p + 65 > - 16$ for $p$ .

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

Subtract $65$ from both sides of the inequality.

$9 p > - 16 - 65$

Subtract $65$ from $- 16$ .

$9 p > - 81$

Divide each term in $9 p > - 81$ by $9$ and simplify.

Divide each term in $9 p > - 81$ by $9$ .

$\frac{9 p}{9} > \frac{- 81}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 p}{9} > \frac{- 81}{9}$

Divide $p$ by $1$ .

$p > \frac{- 81}{9}$

Simplify the right side.

Divide $- 81$ by $9$ .

$p > - 9$

Find the intersection of $p > - 9$ and $p < - 7$ .

$- 9 < p < - 7$

Step 4

Find the union of the solutions.

$- 9 < p < - 5$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 9 < p < - 5$

Interval Notation:

$(- 9, - 5)$

Step 6

Solve the Inequality for p 2-9|p+7|>-16

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