Precalculus Examples

$\frac{5 x (x - y)}{x^{3} (y - x)}$

Step 1

Set the denominator in $\frac{5 x (x - y)}{x^{3} (y - x)}$ equal to $0$ to find where the expression is undefined.

$x^{3} (y - x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x^{3} = 0$

$y - x = 0$

Step 2.2

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.2.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.2.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 2.2.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.2.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.2.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 2.3

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

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Step 2.3.1

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

$y - x = 0$

Step 2.3.2

Solve $y - x = 0$ for $x$ .

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Step 2.3.2.1

Subtract $y$ from both sides of the equation.

$- x = - y$

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.3.2.2.2.2

Divide $x$ by $1$ .

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

Step 2.3.2.2.3

Simplify the right side.

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Step 2.3.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{y}{1}$

Step 2.3.2.2.3.2

Divide $y$ by $1$ .

$x = y$

Step 2.4

The final solution is all the values that make $x^{3} (y - x) = 0$ true.

$x = 0$

$x = y$

$x = 0$

$x = y$

Step 3

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$