Find the Domain (22x+11)/(10x^2+5x)

$\frac{22 x + 11}{10 x^{2} + 5 x}$

Step 1

Set the denominator in $\frac{22 x + 11}{10 x^{2} + 5 x}$ equal to $0$ to find where the expression is undefined.

$10 x^{2} + 5 x = 0$

Step 2

Solve for $x$ .

Factor $5 x$ out of $10 x^{2} + 5 x$ .

Factor $5 x$ out of $10 x^{2}$ .

$5 x (2 x) + 5 x = 0$

Factor $5 x$ out of $5 x$ .

$5 x (2 x) + 5 x (1) = 0$

Factor $5 x$ out of $5 x (2 x) + 5 x (1)$ .

$5 x (2 x + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x + 1 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

The final solution is all the values that make $5 x (2 x + 1) = 0$ true.

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

Step 3

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

Interval Notation:

$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0, - \frac{1}{2}}$

Step 4