Precalculus Examples

$f (x) = \frac{1}{\frac{4}{x - 1} - 2}$

Step 1

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

$x - 1 = 0$

Step 2

Add $1$ to both sides of the equation.

$x = 1$

Step 3

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

$\frac{4}{x - 1} - 2 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Add $2$ to both sides of the equation.

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

Step 4.2

Find the LCD of the terms in the equation.

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Step 4.2.1

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

$x - 1, 1$

Step 4.2.2

Remove parentheses.

$x - 1, 1$

Step 4.2.3

The LCM of one and any expression is the expression.

$x - 1$

Step 4.3

Multiply each term in $\frac{4}{x - 1} = 2$ by $x - 1$ to eliminate the fractions.

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Step 4.3.1

Multiply each term in $\frac{4}{x - 1} = 2$ by $x - 1$ .

$\frac{4}{x - 1} (x - 1) = 2 (x - 1)$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $x - 1$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{4}{x - 1} (x - 1) = 2 (x - 1)$

Step 4.3.2.1.2

Rewrite the expression.

$4 = 2 (x - 1)$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Apply the distributive property.

$4 = 2 x + 2 \cdot - 1$

Step 4.3.3.2

Multiply $2$ by $- 1$ .

$4 = 2 x - 2$

Step 4.4

Solve the equation.

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Step 4.4.1

Rewrite the equation as $2 x - 2 = 4$ .

$2 x - 2 = 4$

Step 4.4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.4.2.1

Add $2$ to both sides of the equation.

$2 x = 4 + 2$

Step 4.4.2.2

Add $4$ and $2$ .

$2 x = 6$

Step 4.4.3

Divide each term in $2 x = 6$ by $2$ and simplify.

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Step 4.4.3.1

Divide each term in $2 x = 6$ by $2$ .

$\frac{2 x}{2} = \frac{6}{2}$

Step 4.4.3.2

Simplify the left side.

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Step 4.4.3.2.1

Cancel the common factor of $2$ .

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Step 4.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{6}{2}$

Step 4.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{2}$

Step 4.4.3.3

Simplify the right side.

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Step 4.4.3.3.1

Divide $6$ by $2$ .

$x = 3$

Step 5

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

Interval Notation:

$(- \infty, 1) \cup (1, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 1, 3}$

Step 6