Algebra Examples

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

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.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Cancel the common factor.

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

Step 4.1.1.2

Rewrite the expression.

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

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $2$ by $2$ .

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

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

Simplify the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Pull terms out from under the radical.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\sqrt{4}$ .

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Step 5.2.2.1.1

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

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

Step 5.2.2.1.2

Pull terms out from under the radical.

$| x - 2 | > | 2 |$

Step 5.2.2.1.3

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

$| x - 2 | > 2$

Step 5.3

Write $| x - 2 | > 2$ as a piecewise.

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Step 5.3.1

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

$x - 2 \geq 0$

Step 5.3.2

Add $2$ to both sides of the inequality.

$x \geq 2$

Step 5.3.3

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

$x - 2 > 2$

Step 5.3.4

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

$x - 2 < 0$

Step 5.3.5

Add $2$ to both sides of the inequality.

$x < 2$

Step 5.3.6

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

$- (x - 2) > 2$

Step 5.3.7

Write as a piecewise.

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

Step 5.3.8

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

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Step 5.3.8.1

Apply the distributive property.

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

Step 5.3.8.2

Multiply $- 1$ by $- 2$ .

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

Step 5.4

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

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Step 5.4.1

Add $2$ to both sides of the inequality.

$x > 2 + 2$

Step 5.4.2

Add $2$ and $2$ .

$x > 4$

Step 5.5

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

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Step 5.5.1

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

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Step 5.5.1.1

Subtract $2$ from both sides of the inequality.

$- x > 2 - 2$

Step 5.5.1.2

Subtract $2$ from $2$ .

$- x > 0$

Step 5.5.2

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

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Step 5.5.2.1

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.

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

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.5.2.2.2

Divide $x$ by $1$ .

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

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Divide $0$ by $- 1$ .

$x < 0$

Step 5.6

Find the union of the solutions.

$x < 0$ or $x > 4$

Step 6