Find the Domain (10/x)/(10/x+1)

$\frac{\frac{10}{x}}{\frac{10}{x} + 1}$

Step 1

Set the denominator in $\frac{10}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2

Set the denominator in $\frac{\frac{10}{x}}{\frac{10}{x} + 1}$ equal to $0$ to find where the expression is undefined.

$\frac{10}{x} + 1 = 0$

Step 3

Solve for $x$ .

Subtract $1$ from both sides of the equation.

$\frac{10}{x} = - 1$

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.

$x, 1$

The LCM of one and any expression is the expression.

$x$

Multiply each term in $\frac{10}{x} = - 1$ by $x$ to eliminate the fractions.

Multiply each term in $\frac{10}{x} = - 1$ by $x$ .

$\frac{10}{x} x = - x$

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{10}{x} x = - x$

Rewrite the expression.

$10 = - x$

Solve the equation.

Rewrite the equation as $- x = 10$ .

$- x = 10$

Divide each term in $- x = 10$ by $- 1$ and simplify.

Divide each term in $- x = 10$ by $- 1$ .

$\frac{- x}{- 1} = \frac{10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{10}{- 1}$

Divide $x$ by $1$ .

$x = \frac{10}{- 1}$

Simplify the right side.

Divide $10$ by $- 1$ .

$x = - 10$

Step 4

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

Interval Notation:

$(- \infty, - 10) \cup (- 10, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0, - 10}$

Step 5

Find the Domain (10/x)/(10/x+1)

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