Graph 1/2(x-2)^2-2>0

$\frac{1}{2} {(x - 2)}^{2} - 2 > 0$

Step 1

Add $2$ to both sides of the inequality.

$\frac{1}{2} \cdot {(x - 2)}^{2} > 2$

Step 2

Combine $\frac{1}{2}$ and ${(x - 2)}^{2}$ .

$\frac{{(x - 2)}^{2}}{2} > 2$

Step 3

Multiply both sides by $2$ .

$\frac{{(x - 2)}^{2}}{2} \cdot 2 > 2 \cdot 2$

Step 4

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{{(x - 2)}^{2}}{2} \cdot 2 > 2 \cdot 2$

Rewrite the expression.

${(x - 2)}^{2} > 2 \cdot 2$

Simplify the right side.

Multiply $2$ by $2$ .

${(x - 2)}^{2} > 4$

Step 5

Solve for $x$ .

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

$\sqrt{{(x - 2)}^{2}} > \sqrt{4}$

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

$| x - 2 | > \sqrt{4}$

Simplify the right side.

Simplify $\sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$| x - 2 | > | 2 |$

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| x - 2 | > 2$

Write $| x - 2 | > 2$ 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.

$x - 2 > 2$

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 $- 1$ .

$- (x - 2) > 2$

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

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

Add $2$ to both sides of the inequality.

$x > 2 + 2$

Add $2$ and $2$ .

$x > 4$

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

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

Subtract $2$ from both sides of the inequality.

$- x > 2 - 2$

Subtract $2$ from $2$ .

$- x > 0$

Divide each term in $- x > 0$ by $- 1$ and simplify.

Divide each term in $- x > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{0}{- 1}$

Divide $x$ by $1$ .

$x < \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x < 0$

Find the union of the solutions.

$x < 0$ or $x > 4$

Step 6

Graph 1/2(x-2)^2-2>0

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