Algebra Examples

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

Step 1

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

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

Step 2

Simplify the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Pull terms out from under the radical.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $\sqrt{0}$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Pull terms out from under the radical.

$| x + 2 | > | 0 |$

Step 2.2.1.3

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

$| x + 2 | > 0$

Step 3

Write $| x + 2 | > 0$ as a piecewise.

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Step 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 3.2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 3.3

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

$x + 2 > 0$

Step 3.4

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

$x + 2 < 0$

Step 3.5

Subtract $2$ from both sides of the inequality.

$x < - 2$

Step 3.6

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

$- (x + 2) > 0$

Step 3.7

Write as a piecewise.

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

Step 3.8

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

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Step 3.8.1

Apply the distributive property.

${\begin{cases} x + 2 > 0 & x \geq - 2 \\ - x - 1 \cdot 2 > 0 & x < - 2 \end{cases}$

Step 3.8.2

Multiply $- 1$ by $2$ .

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

Step 4

Subtract $2$ from both sides of the inequality.

$x > - 2$

Step 5

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

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Step 5.1

Add $2$ to both sides of the inequality.

$- x > 2$

Step 5.2

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

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Step 5.2.1

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.

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $x$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $2$ by $- 1$ .

$x < - 2$

Step 6

Find the union of the solutions.

$x < - 2$ or $x > - 2$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < - 2 or x > - 2$

Interval Notation:

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

Step 8