Solve the Inequality for x (x-2)^2>0

${(x - 2)}^{2} > 0$

Take the square root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{{(x - 2)}^{2}} > \sqrt{0}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x - 2 | > \sqrt{0}$

Simplify the right side.

Simplify $\sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$| x - 2 | > \sqrt{0^{2}}$

Pull terms out from under the radical.

$| x - 2 | > | 0 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x - 2 | > 0$

Write $| x - 2 | > 0$ 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.

$x - 2 > 0$

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 $- 1$ .

$- (x - 2) > 0$

Write as a piecewise.

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

Simplify $- (x - 2) > 0$ .

Apply the distributive property.

${\begin{cases} x - 2 > 0 & x \geq 2 \\ - x - - 2 > 0 & x < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} x - 2 > 0 & x \geq 2 \\ - x + 2 > 0 & x < 2 \end{cases}$

Add $2$ to both sides of the inequality.

$x > 2$

Solve $- x + 2 > 0$ for $x$ .

Subtract $2$ from both sides of the inequality.

$- x > - 2$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 2$ by $- 1$ .

$x < 2$

Find the union of the solutions.

$x < 2$ or $x > 2$

The result can be shown in multiple forms.

Inequality Form:

$x < 2 or x > 2$

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Solve the Inequality for x (x-2)^2>0

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