Solve the Absolute Value Inequality for x |2x-8|>12

$| 2 x - 8 | > 12$

Write $| 2 x - 8 | > 12$ as a piecewise.

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

$2 x - 8 \geq 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$2 x \geq 8$

Divide each term in $2 x \geq 8$ by $2$ and simplify.

Divide each term in $2 x \geq 8$ by $2$ .

$\frac{2 x}{2} \geq \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{8}{2}$

Divide $x$ by $1$ .

$x \geq \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$x \geq 4$

In the piece where $2 x - 8$ is non-negative, remove the absolute value.

$2 x - 8 > 12$

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

$2 x - 8 < 0$

Solve the inequality.

Add $8$ to both sides of the inequality.

$2 x < 8$

Divide each term in $2 x < 8$ by $2$ and simplify.

Divide each term in $2 x < 8$ by $2$ .

$\frac{2 x}{2} < \frac{8}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} < \frac{8}{2}$

Divide $x$ by $1$ .

$x < \frac{8}{2}$

Simplify the right side.

Divide $8$ by $2$ .

$x < 4$

In the piece where $2 x - 8$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 8) > 12$

Write as a piecewise.

${\begin{cases} 2 x - 8 > 12 & x \geq 4 \\ - (2 x - 8) > 12 & x < 4 \end{cases}$

Simplify $- (2 x - 8) > 12$ .

Apply the distributive property.

${\begin{cases} 2 x - 8 > 12 & x \geq 4 \\ - (2 x) - - 8 > 12 & x < 4 \end{cases}$

Multiply $2$ by $- 1$ .

${\begin{cases} 2 x - 8 > 12 & x \geq 4 \\ - 2 x - - 8 > 12 & x < 4 \end{cases}$

Multiply $- 1$ by $- 8$ .

${\begin{cases} 2 x - 8 > 12 & x \geq 4 \\ - 2 x + 8 > 12 & x < 4 \end{cases}$

Solve $2 x - 8 > 12$ for $x$ .

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

Add $8$ to both sides of the inequality.

$2 x > 12 + 8$

Add $12$ and $8$ .

$2 x > 20$

Divide each term in $2 x > 20$ by $2$ and simplify.

Divide each term in $2 x > 20$ by $2$ .

$\frac{2 x}{2} > \frac{20}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{20}{2}$

Divide $x$ by $1$ .

$x > \frac{20}{2}$

Simplify the right side.

Divide $20$ by $2$ .

$x > 10$

Solve $- 2 x + 8 > 12$ for $x$ .

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

Subtract $8$ from both sides of the inequality.

$- 2 x > 12 - 8$

Subtract $8$ from $12$ .

$- 2 x > 4$

Divide each term in $- 2 x > 4$ by $- 2$ and simplify.

Divide each term in $- 2 x > 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{4}{- 2}$

Divide $x$ by $1$ .

$x < \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$x < - 2$

Find the union of the solutions.

$x < - 2$ or $x > 10$

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 10$

Interval Notation:

$(- \infty, - 2) \cup (10, \infty)$

Solve the Absolute Value Inequality for x |2x-8|>12

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