Solve for x |x-2|-3<=0

$| x - 2 | - 3 \leq 0$

Write $| x - 2 | - 3 \leq 0$ as a piecewise.

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

$x - 2 \geq 0$

Add $2$ to both sides of the inequality.

$x \geq 2$

In the piece where $x - 2$ is non-negative, remove the absolute value.

$x - 2 - 3 \leq 0$

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

$x - 2 < 0$

Add $2$ to both sides of the inequality.

$x < 2$

In the piece where $x - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x - 2) - 3 \leq 0$

Write as a piecewise.

${\begin{cases} x - 2 - 3 \leq 0 & x \geq 2 \\ - (x - 2) - 3 \leq 0 & x < 2 \end{cases}$

Subtract $3$ from $- 2$ .

${\begin{cases} x - 5 \leq 0 & x \geq 2 \\ - (x - 2) - 3 \leq 0 & x < 2 \end{cases}$

Simplify $- (x - 2) - 3 \leq 0$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} x - 5 \leq 0 & x \geq 2 \\ - x - - 2 - 3 \leq 0 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} x - 5 \leq 0 & x \geq 2 \\ - x + 2 - 3 \leq 0 & x < 2 \end{cases}$

Subtract $3$ from $2$ .

${\begin{cases} x - 5 \leq 0 & x \geq 2 \\ - x - 1 \leq 0 & x < 2 \end{cases}$

Solve $x - 5 \leq 0$ when $x \geq 2$ .

Add $5$ to both sides of the inequality.

$x \leq 5$

Find the intersection of $x \leq 5$ and $x \geq 2$ .

$2 \leq x \leq 5$

Solve $- x - 1 \leq 0$ when $x < 2$ .

Solve $- x - 1 \leq 0$ for $x$ .

Add $1$ to both sides of the inequality.

$- x \leq 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $1$ by $- 1$ .

$x \geq - 1$

Find the intersection of $x \geq - 1$ and $x < 2$ .

$- 1 \leq x < 2$

Find the union of the solutions.

$- 1 \leq x \leq 5$

The result can be shown in multiple forms.

Inequality Form:

$- 1 \leq x \leq 5$

Interval Notation:

$[- 1, 5]$

Solve for x |x-2|-3<=0

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