Solve the Inequality for x |x+4|-8>2

$| x + 4 | - 8 > 2$

Step 1

Write $| x + 4 | - 8 > 2$ as a piecewise.

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

$x + 4 \geq 0$

Subtract $4$ from both sides of the inequality.

$x \geq - 4$

In the piece where $x + 4$ is non-negative, remove the absolute value.

$x + 4 - 8 > 2$

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

$x + 4 < 0$

Subtract $4$ from both sides of the inequality.

$x < - 4$

In the piece where $x + 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x + 4) - 8 > 2$

Write as a piecewise.

${\begin{cases} x + 4 - 8 > 2 & x \geq - 4 \\ - (x + 4) - 8 > 2 & x < - 4 \end{cases}$

Subtract $8$ from $4$ .

${\begin{cases} x - 4 > 2 & x \geq - 4 \\ - (x + 4) - 8 > 2 & x < - 4 \end{cases}$

Simplify $- (x + 4) - 8 > 2$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x - 4 > 2 & x \geq - 4 \\ - x - 1 \cdot 4 - 8 > 2 & x < - 4 \end{cases}$

Multiply $- 1$ by $4$ .

${\begin{cases} x - 4 > 2 & x \geq - 4 \\ - x - 4 - 8 > 2 & x < - 4 \end{cases}$

Subtract $8$ from $- 4$ .

${\begin{cases} x - 4 > 2 & x \geq - 4 \\ - x - 12 > 2 & x < - 4 \end{cases}$

Step 2

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

Add $4$ to both sides of the inequality.

$x > 2 + 4$

Add $2$ and $4$ .

$x > 6$

Step 3

Solve $- x - 12 > 2$ for $x$ .

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

Add $12$ to both sides of the inequality.

$- x > 2 + 12$

Add $2$ and $12$ .

$- x > 14$

Divide each term in $- x > 14$ by $- 1$ and simplify.

Divide each term in $- x > 14$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{14}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{14}{- 1}$

Divide $x$ by $1$ .

$x < \frac{14}{- 1}$

Simplify the right side.

Divide $14$ by $- 1$ .

$x < - 14$

Step 4

Find the union of the solutions.

$x < - 14$ or $x > 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 14 or x > 6$

Interval Notation:

$(- \infty, - 14) \cup (6, \infty)$

Step 6

Solve the Inequality for x |x+4|-8>2

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