Solve the Inequality for x |x-5|>=0

$| x - 5 | \geq 0$

Write $| x - 5 | \geq 0$ as a piecewise.

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

$x - 5 \geq 0$

Add $5$ to both sides of the inequality.

$x \geq 5$

In the piece where $x - 5$ is non-negative, remove the absolute value.

$x - 5 \geq 0$

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

$x - 5 < 0$

Add $5$ to both sides of the inequality.

$x < 5$

In the piece where $x - 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 5) \geq 0$

Write as a piecewise.

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

Simplify $- (x - 5) \geq 0$ .

Apply the distributive property.

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

Multiply $- 1$ by $- 5$ .

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

Add $5$ to both sides of the inequality.

$x \geq 5$

Solve $- x + 5 \geq 0$ when $x < 5$ .

Solve $- x + 5 \geq 0$ for $x$ .

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x \leq 5$

Find the intersection of $x \leq 5$ and $x < 5$ .

$x < 5$

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-5|>=0

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