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

$| x - 8 | > 4$

Step 1

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

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

$x - 8 \geq 0$

Add $8$ to both sides of the inequality.

$x \geq 8$

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

$x - 8 > 4$

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

$x - 8 < 0$

Add $8$ to both sides of the inequality.

$x < 8$

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

$- (x - 8) > 4$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 8$ .

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

Step 2

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

Add $8$ to both sides of the inequality.

$x > 4 + 8$

Add $4$ and $8$ .

$x > 12$

Step 3

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

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

Subtract $8$ from both sides of the inequality.

$- x > 4 - 8$

Subtract $8$ from $4$ .

$- x > - 4$

Divide each term in $- x > - 4$ by $- 1$ and simplify.

Divide each term in $- x > - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x < \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x < 4$

Step 4

Find the union of the solutions.

$x < 4$ or $x > 12$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < 4 or x > 12$

Interval Notation:

$(- \infty, 4) \cup (12, \infty)$

Step 6

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

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