Precalculus Examples

$\frac{(\frac{x}{1 + x})}{1 + (\frac{x}{1 + x})}$

Step 1

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

$1 + x = 0$

Step 2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 3

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

$1 + \frac{x}{1 + x} = 0$

Step 4

Solve for $x$ .

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Step 4.1

Find the LCD of the terms in the equation.

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Step 4.1.1

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

$1, 1 + x, 1$

Step 4.1.2

Remove parentheses.

$1, 1 + x, 1$

Step 4.1.3

The LCM of one and any expression is the expression.

$1 + x$

Step 4.2

Multiply each term in $1 + \frac{x}{1 + x} = 0$ by $1 + x$ to eliminate the fractions.

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Step 4.2.1

Multiply each term in $1 + \frac{x}{1 + x} = 0$ by $1 + x$ .

$1 (1 + x) + \frac{x}{1 + x} (1 + x) = 0 (1 + x)$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Multiply $1 + x$ by $1$ .

$1 + x + \frac{x}{1 + x} (1 + x) = 0 (1 + x)$

Step 4.2.2.1.2

Cancel the common factor of $1 + x$ .

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Step 4.2.2.1.2.1

Cancel the common factor.

$1 + x + \frac{x}{1 + x} (1 + x) = 0 (1 + x)$

Step 4.2.2.1.2.2

Rewrite the expression.

$1 + x + x = 0 (1 + x)$

Step 4.2.2.2

Add $x$ and $x$ .

$1 + 2 x = 0 (1 + x)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Multiply $0$ by $1 + x$ .

$1 + 2 x = 0$

Step 4.3

Solve the equation.

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 4.3.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 4.3.2.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Cancel the common factor of $2$ .

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Step 4.3.2.2.1.1

Cancel the common factor.

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

Step 4.3.2.2.1.2

Divide $x$ by $1$ .

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

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Move the negative in front of the fraction.

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

Step 5

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

Interval Notation:

$(- \infty, - 1) \cup (- 1, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{1}{2}, - 1}$

Step 6