Solve the Absolute Value Inequality for x |2x-4|>8

$| 2 x - 4 | > 8$

Step 1

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

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

$2 x - 4 \geq 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$2 x \geq 4$

Divide each term in $2 x \geq 4$ by $2$ and simplify.

Divide each term in $2 x \geq 4$ by $2$ .

$\frac{2 x}{2} \geq \frac{4}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{4}{2}$

Divide $x$ by $1$ .

$x \geq \frac{4}{2}$

Simplify the right side.

Divide $4$ by $2$ .

$x \geq 2$

In the piece where $2 x - 4$ is non-negative, remove the absolute value.

$2 x - 4 > 8$

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

$2 x - 4 < 0$

Solve the inequality.

Add $4$ to both sides of the inequality.

$2 x < 4$

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

Divide each term in $2 x < 4$ by $2$ .

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

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 $2$ .

$x < 2$

In the piece where $2 x - 4$ is negative, remove the absolute value and multiply by $- 1$ .

$- (2 x - 4) > 8$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 4$ .

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

Step 2

Solve $2 x - 4 > 8$ for $x$ .

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

Add $4$ to both sides of the inequality.

$2 x > 8 + 4$

Add $8$ and $4$ .

$2 x > 12$

Divide each term in $2 x > 12$ by $2$ and simplify.

Divide each term in $2 x > 12$ by $2$ .

$\frac{2 x}{2} > \frac{12}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} > \frac{12}{2}$

Divide $x$ by $1$ .

$x > \frac{12}{2}$

Simplify the right side.

Divide $12$ by $2$ .

$x > 6$

Step 3

Solve $- 2 x + 4 > 8$ for $x$ .

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

Subtract $4$ from both sides of the inequality.

$- 2 x > 8 - 4$

Subtract $4$ from $8$ .

$- 2 x > 4$

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 $- 2$ .

$x < - 2$

Step 4

Find the union of the solutions.

$x < - 2$ or $x > 6$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > 6$

Interval Notation:

$(- \infty, - 2) \cup (6, \infty)$

Step 6

Solve the Absolute Value Inequality for x |2x-4|>8

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