Precalculus Examples

$\frac{1}{x^{2} - x y} - \frac{y^{2}}{x^{2} - y^{2}} \cdot \frac{x + y}{y^{3}}$

Step 1

Set the denominator in $\frac{1}{x^{2} - x y}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

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

$x \cdot x - x y = 0$

Step 2.1.2

Factor $x$ out of $- x y$ .

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

Step 2.1.3

Factor $x$ out of $x \cdot x + x (- 1 y)$ .

$x (x - 1 y) = 0$

Step 2.2

Rewrite $- 1 y$ as $- y$ .

$x (x - y) = 0$

Step 2.3

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

$x = 0$

$x - y = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

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

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Step 2.5.1

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

$x - y = 0$

Step 2.5.2

Add $y$ to both sides of the equation.

$x = y$

Step 2.6

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

$x = 0$

$x = y$

$x = 0$

$x = y$

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

Add $y^{2}$ to both sides of the equation.

$x^{2} = y^{2}$

Step 4.2

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| x | = | y |$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x = \pm | y |$

Step 4.3.2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.3.2.1

First, use the positive value of the $\pm$ to find the first solution.

$x = | y |$

Step 4.3.2.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - | y |$

Step 4.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

$x = | y |$

$x = - | y |$

Step 5

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

$y^{3} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

$y = 0$

Step 7

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$