Pre-Algebra Examples

$- 2 x^{2} + 4 > 0$

Step 1

Subtract $4$ from both sides of the inequality.

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

Step 2

Divide each term in $- 2 x^{2} > - 4$ by $- 2$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- 2 x^{2} > - 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x^{2}}{- 2} < \frac{- 4}{- 2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{- 2 x^{2}}{- 2} < \frac{- 4}{- 2}$

Step 2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} < \frac{- 4}{- 2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Divide $- 4$ by $- 2$ .

$x^{2} < 2$

Step 3

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

$\sqrt{x^{2}} < \sqrt{2}$

Step 4

Simplify the left side.

Tap for more steps...

Step 4.1

Pull terms out from under the radical.

$| x | < \sqrt{2}$

Step 5

Write $| x | < \sqrt{2}$ as a piecewise.

Tap for more steps...

Step 5.1

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

$x \geq 0$

Step 5.2

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

$x < \sqrt{2}$

Step 5.3

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

$x < 0$

Step 5.4

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

$- x < \sqrt{2}$

Step 5.5

Write as a piecewise.

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

Step 6

Find the intersection of $x < \sqrt{2}$ and $x \geq 0$ .

$0 \leq x < \sqrt{2}$

Step 7

Solve $- x < \sqrt{2}$ when $x < 0$ .

Tap for more steps...

Step 7.1

Divide each term in $- x < \sqrt{2}$ by $- 1$ and simplify.

Tap for more steps...

Step 7.1.1

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

$\frac{- x}{- 1} > \frac{\sqrt{2}}{- 1}$

Step 7.1.2

Simplify the left side.

Tap for more steps...

Step 7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{\sqrt{2}}{- 1}$

Step 7.1.2.2

Divide $x$ by $1$ .

$x > \frac{\sqrt{2}}{- 1}$

Step 7.1.3

Simplify the right side.

Tap for more steps...

Step 7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{2}}{- 1}$ .

$x > - 1 \cdot \sqrt{2}$

Step 7.1.3.2

Rewrite $- 1 \cdot \sqrt{2}$ as $- \sqrt{2}$ .

$x > - \sqrt{2}$

Step 7.2

Find the intersection of $x > - \sqrt{2}$ and $x < 0$ .

$- \sqrt{2} < x < 0$

Step 8

Find the union of the solutions.

$- \sqrt{2} < x < \sqrt{2}$

Step 9