Solve the Inequality for x |x+8|>5

$| x + 8 | > 5$

Write $| x + 8 | > 5$ as a piecewise.

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

$x + 8 \geq 0$

Subtract $8$ from both sides of the inequality.

$x \geq - 8$

In the piece where $x + 8$ is non-negative, remove the absolute value.

$x + 8 > 5$

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

$x + 8 < 0$

Subtract $8$ from both sides of the inequality.

$x < - 8$

In the piece where $x + 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 8) > 5$

Write as a piecewise.

${\begin{cases} x + 8 > 5 & x \geq - 8 \\ - (x + 8) > 5 & x < - 8 \end{cases}$

Simplify $- (x + 8) > 5$ .

Apply the distributive property.

${\begin{cases} x + 8 > 5 & x \geq - 8 \\ - x - 1 \cdot 8 > 5 & x < - 8 \end{cases}$

Multiply $- 1$ by $8$ .

${\begin{cases} x + 8 > 5 & x \geq - 8 \\ - x - 8 > 5 & x < - 8 \end{cases}$

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

Subtract $8$ from both sides of the inequality.

$x > 5 - 8$

Subtract $8$ from $5$ .

$x > - 3$

Solve $- x - 8 > 5$ for $x$ .

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

Add $8$ to both sides of the inequality.

$- x > 5 + 8$

Add $5$ and $8$ .

$- x > 13$

Divide each term in $- x > 13$ by $- 1$ and simplify.

Divide each term in $- x > 13$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{13}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{13}{- 1}$

Divide $x$ by $1$ .

$x < \frac{13}{- 1}$

Simplify the right side.

Divide $13$ by $- 1$ .

$x < - 13$

Find the union of the solutions.

$x < - 13$ or $x > - 3$

The result can be shown in multiple forms.

Inequality Form:

$x < - 13 or x > - 3$

Interval Notation:

$(- \infty, - 13) \cup (- 3, \infty)$

Solve the Inequality for x |x+8|>5

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