Solve the Inequality for x |x|>x-1

$| x | > x - 1$

Write $| x | > x - 1$ 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.

$x > x - 1$

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 $- 1$ .

$- x > x - 1$

Write as a piecewise.

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

Solve $x > x - 1$ when $x \geq 0$ .

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

Move all terms containing $x$ to the left side of the inequality.

Subtract $x$ from both sides of the inequality.

$x - x > - 1$

Subtract $x$ from $x$ .

$0 > - 1$

Since $0 > - 1$ , the inequality will always be true.

Always true

Find the intersection.

$x \geq 0$

Solve $- x > x - 1$ when $x < 0$ .

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

Move all terms containing $x$ to the left side of the inequality.

Subtract $x$ from both sides of the inequality.

$- x - x > - 1$

Subtract $x$ from $- x$ .

$- 2 x > - 1$

Divide each term in $- 2 x > - 1$ by $- 2$ and simplify.

Divide each term in $- 2 x > - 1$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{- 1}{- 2}$

Divide $x$ by $1$ .

$x < \frac{- 1}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x < \frac{1}{2}$

Find the intersection of $x < \frac{1}{2}$ and $x < 0$ .

$x < 0$

Find the union of the solutions.

All real numbers

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve the Inequality for x |x|>x-1

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