Solve the Absolute Value Inequality for x |x|>5

$| x | > 5$

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

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

$x \geq 0$

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

$x > 5$

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

$x < 0$

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

$- x > 5$

Write as a piecewise.

${\begin{cases} x > 5 & x \geq 0 \\ - x > 5 & x < 0 \end{cases}$

Find the intersection of $x > 5$ and $x \geq 0$ .

$x > 5$

Divide each term in $- x > 5$ by $- 1$ and simplify.

Divide each term in $- x > 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{5}{- 1}$

Divide $x$ by $1$ .

$x < \frac{5}{- 1}$

Simplify the right side.

Divide $5$ by $- 1$ .

$x < - 5$

Find the union of the solutions.

$x < - 5$ or $x > 5$

The result can be shown in multiple forms.

Inequality Form:

$x < - 5 or x > 5$

Interval Notation:

$(- \infty, - 5) \cup (5, \infty)$

Solve the Absolute Value Inequality for x |x|>5

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