Find the Domain (5-x)/(x^2-4x)

$\frac{5 - x}{x^{2} - 4 x}$

Step 1

Set the denominator in $\frac{5 - x}{x^{2} - 4 x}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 x = 0$

Step 2

Solve for $x$ .

Factor $x$ out of $x^{2} - 4 x$ .

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

$x \cdot x - 4 x = 0$

Factor $x$ out of $- 4 x$ .

$x \cdot x + x \cdot - 4 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 4$ .

$x (x - 4) = 0$

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

$x = 0$

$x - 4 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $x (x - 4) = 0$ true.

$x = 0, 4$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, 4) \cup (4, \infty)$

Set-Builder Notation:

${x | x \neq 0, 4}$

Step 4