Precalculus Examples

$\frac{2 - \frac{6}{x}}{1 - \frac{4}{x}}$

Step 1

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

$x = 0$

Step 2

Set the denominator in $\frac{2 - \frac{6}{x}}{1 - \frac{4}{x}}$ equal to $0$ to find where the expression is undefined.

$1 - \frac{4}{x} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $1$ from both sides of the equation.

$- \frac{4}{x} = - 1$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$x$

Step 3.3

Multiply each term in $- \frac{4}{x} = - 1$ by $x$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{4}{x} = - 1$ by $x$ .

$- \frac{4}{x} x = - x$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $x$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{4}{x}$ into the numerator.

$\frac{- 4}{x} x = - x$

Step 3.3.2.1.2

Cancel the common factor.

$\frac{- 4}{x} x = - x$

Step 3.3.2.1.3

Rewrite the expression.

$- 4 = - x$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $- x = - 4$ .

$- x = - 4$

Step 3.4.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

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Step 3.4.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 3.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 4

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

Interval Notation:

$(- \infty, 0) \cup (0, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 0, 4}$

Step 5