Find the Domain ((x+1)/(x-6)+1)/((x+1)/(x-6)-6)

$\frac{\frac{x + 1}{x - 6} + 1}{\frac{x + 1}{x - 6} - 6}$

Step 1

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

$x - 6 = 0$

Step 2

Add $6$ to both sides of the equation.

$x = 6$

Step 3

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

$\frac{x + 1}{x - 6} - 6 = 0$

Step 4

Solve for $x$ .

Find the LCD of the terms in the equation.

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

$x - 6, 1, 1$

Remove parentheses.

$x - 6, 1, 1$

The LCM of one and any expression is the expression.

$x - 6$

Multiply each term in $\frac{x + 1}{x - 6} - 6 = 0$ by $x - 6$ to eliminate the fractions.

Multiply each term in $\frac{x + 1}{x - 6} - 6 = 0$ by $x - 6$ .

$\frac{x + 1}{x - 6} (x - 6) - 6 (x - 6) = 0 (x - 6)$

Simplify the left side.

Simplify each term.

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{x + 1}{x - 6} (x - 6) - 6 (x - 6) = 0 (x - 6)$

Rewrite the expression.

$x + 1 - 6 (x - 6) = 0 (x - 6)$

Apply the distributive property.

$x + 1 - 6 x - 6 \cdot - 6 = 0 (x - 6)$

Multiply $- 6$ by $- 6$ .

$x + 1 - 6 x + 36 = 0 (x - 6)$

Simplify by adding terms.

Subtract $6 x$ from $x$ .

$- 5 x + 1 + 36 = 0 (x - 6)$

Add $1$ and $36$ .

$- 5 x + 37 = 0 (x - 6)$

Simplify the right side.

Multiply $0$ by $x - 6$ .

$- 5 x + 37 = 0$

Solve the equation.

Subtract $37$ from both sides of the equation.

$- 5 x = - 37$

Divide each term in $- 5 x = - 37$ by $- 5$ and simplify.

Divide each term in $- 5 x = - 37$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 37}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 37}{- 5}$

Divide $x$ by $1$ .

$x = \frac{- 37}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{37}{5}$

Step 5

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

Interval Notation:

$(- \infty, 6) \cup (6, \frac{37}{5}) \cup (\frac{37}{5}, \infty)$

Set-Builder Notation:

${x | x \neq 6, \frac{37}{5}}$

Step 6

Find the Domain ((x+1)/(x-6)+1)/((x+1)/(x-6)-6)

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