Precalculus Examples

$f (x) = \frac{x - 4}{6 - 3 x} + \sqrt{6 - 2 x}$

Step 1

Set the radicand in $\sqrt{6 - 2 x}$ greater than or equal to $0$ to find where the expression is defined.

$6 - 2 x \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6$ from both sides of the inequality.

$- 2 x \geq - 6$

Step 2.2

Divide each term in $- 2 x \geq - 6$ by $- 2$ and simplify.

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Step 2.2.1

Divide each term in $- 2 x \geq - 6$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 x}{- 2} \leq \frac{- 6}{- 2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} \leq \frac{- 6}{- 2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $- 6$ by $- 2$ .

$x \leq 3$

Step 3

Set the denominator in $\frac{x - 4}{6 - 3 x}$ equal to $0$ to find where the expression is undefined.

$6 - 3 x = 0$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $6$ from both sides of the equation.

$- 3 x = - 6$

Step 4.2

Divide each term in $- 3 x = - 6$ by $- 3$ and simplify.

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Step 4.2.1

Divide each term in $- 3 x = - 6$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 6}{- 3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 6}{- 3}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $- 6$ by $- 3$ .

$x = 2$

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${x | x \leq 3, x \neq 2}$

Step 6