Solve the Inequality for x |x-2|>3

$| x - 2 | > 3$

Write $| x - 2 | > 3$ 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.

$x - 2 > 3$

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 $- 1$ .

$- (x - 2) > 3$

Write as a piecewise.

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

Simplify $- (x - 2) > 3$ .

Apply the distributive property.

${\begin{cases} x - 2 > 3 & x \geq 2 \\ - x - - 2 > 3 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} x - 2 > 3 & x \geq 2 \\ - x + 2 > 3 & x < 2 \end{cases}$

Move all terms not containing $x$ to the right side of the inequality.

Add $2$ to both sides of the inequality.

$x > 3 + 2$

Add $3$ and $2$ .

$x > 5$

Solve $- x + 2 > 3$ for $x$ .

Move all terms not containing $x$ to the right side of the inequality.

Subtract $2$ from both sides of the inequality.

$- x > 3 - 2$

Subtract $2$ from $3$ .

$- x > 1$

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 $- 1$ .

$x < - 1$

Find the union of the solutions.

$x < - 1$ or $x > 5$

The result can be shown in multiple forms.

Inequality Form:

$x < - 1 or x > 5$

Interval Notation:

$(- \infty, - 1) \cup (5, \infty)$

Solve the Inequality for x |x-2|>3

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