Solve the Absolute Value Inequality for x |x|-1>0

$| x | - 1 > 0$

Step 1

Write $| x | - 1 > 0$ 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.

$x - 1 > 0$

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 $- 1$ .

$- x - 1 > 0$

Write as a piecewise.

${\begin{cases} x - 1 > 0 & x \geq 0 \\ - x - 1 > 0 & x < 0 \end{cases}$

Step 2

Add $1$ to both sides of the inequality.

$x > 1$

Step 3

Solve $- x - 1 > 0$ for $x$ .

Add $1$ to both sides of the inequality.

$- x > 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x < - 1$

Step 4

Find the union of the solutions.

$x < - 1$ or $x > 1$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > 1$

Interval Notation:

$(- \infty, - 1) \cup (1, \infty)$

Step 6

Solve the Absolute Value Inequality for x |x|-1>0

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