Trigonometry Examples

$x^{2} - 3 < 0$

Step 1

Add $3$ to both sides of the inequality.

$x^{2} < 3$

Step 2

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

$\sqrt{x^{2}} < \sqrt{3}$

Step 3

Simplify the left side.

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Step 3.1

Pull terms out from under the radical.

$| x | < \sqrt{3}$

Step 4

Write $| x | < \sqrt{3}$ as a piecewise.

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Step 4.1

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

$x \geq 0$

Step 4.2

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

$x < \sqrt{3}$

Step 4.3

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

$x < 0$

Step 4.4

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

$- x < \sqrt{3}$

Step 4.5

Write as a piecewise.

${\begin{cases} x < \sqrt{3} & x \geq 0 \\ - x < \sqrt{3} & x < 0 \end{cases}$

Step 5

Find the intersection of $x < \sqrt{3}$ and $x \geq 0$ .

$0 \leq x < \sqrt{3}$

Step 6

Solve $- x < \sqrt{3}$ when $x < 0$ .

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Step 6.1

Divide each term in $- x < \sqrt{3}$ by $- 1$ and simplify.

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Step 6.1.1

Divide each term in $- x < \sqrt{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} > \frac{\sqrt{3}}{- 1}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} > \frac{\sqrt{3}}{- 1}$

Step 6.1.2.2

Divide $x$ by $1$ .

$x > \frac{\sqrt{3}}{- 1}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$x > - 1 \cdot \sqrt{3}$

Step 6.1.3.2

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$x > - \sqrt{3}$

Step 6.2

Find the intersection of $x > - \sqrt{3}$ and $x < 0$ .

$- \sqrt{3} < x < 0$

Step 7

Find the union of the solutions.

$- \sqrt{3} < x < \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- \sqrt{3} < x < \sqrt{3}$

Interval Notation:

$(- \sqrt{3}, \sqrt{3})$

Step 9