Precalculus Examples

$\frac{\sqrt{- 3 x}}{x^{2} - 1}$

Step 1

Set the radicand in $\sqrt{- 3 x}$ greater than or equal to $0$ to find where the expression is defined.

$- 3 x \geq 0$

Step 2

Divide each term in $- 3 x \geq 0$ by $- 3$ and simplify.

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Step 2.1

Divide each term in $- 3 x \geq 0$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 x}{- 3} \leq \frac{0}{- 3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} \leq \frac{0}{- 3}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{0}{- 3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 3$ .

$x \leq 0$

Step 3

Set the denominator in $\frac{\sqrt{- 3 x}}{x^{2} - 1}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 1 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $1$ to both sides of the equation.

$x^{2} = 1$

Step 4.2

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

$x = \pm \sqrt{1}$

Step 4.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 5

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 1) \cup (- 1, 0]$

Set-Builder Notation:

${x | x \leq 0, x \neq - 1}$

Step 6