Precalculus Examples

$\frac{\sqrt{2 + x}}{3 - x}$

Step 1

Set the radicand in $\sqrt{2 + x}$ greater than or equal to $0$ to find where the expression is defined.

$2 + x \geq 0$

Step 2

Subtract $2$ from both sides of the inequality.

$x \geq - 2$

Step 3

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

$3 - x = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $3$ from both sides of the equation.

$- x = - 3$

Step 4.2

Divide each term in $- x = - 3$ by $- 1$ and simplify.

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Step 4.2.1

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Step 4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 3$ by $- 1$ .

$x = 3$

Step 5

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

Interval Notation:

$[- 2, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \geq - 2, x \neq 3}$

Step 6