Precalculus Examples

$\frac{{(x - y)}^{2}}{2 x y + 6 y} \cdot \frac{4 x + 12}{x^{2} - y^{2}}$

Step 1

Set the denominator in $\frac{{(x - y)}^{2}}{2 x y + 6 y}$ equal to $0$ to find where the expression is undefined.

$2 x y + 6 y = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $6 y$ from both sides of the equation.

$2 x y = - 6 y$

Step 2.2

Divide each term in $2 x y = - 6 y$ by $2 y$ and simplify.

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Step 2.2.1

Divide each term in $2 x y = - 6 y$ by $2 y$ .

$\frac{2 x y}{2 y} = \frac{- 6 y}{2 y}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x y}{2 y} = \frac{- 6 y}{2 y}$

Step 2.2.2.1.2

Rewrite the expression.

$\frac{x y}{y} = \frac{- 6 y}{2 y}$

Step 2.2.2.2

Cancel the common factor of $y$ .

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Step 2.2.2.2.1

Cancel the common factor.

$\frac{x y}{y} = \frac{- 6 y}{2 y}$

Step 2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6 y}{2 y}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $- 6$ and $2$ .

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Step 2.2.3.1.1

Factor $2$ out of $- 6 y$ .

$x = \frac{2 (- 3 y)}{2 y}$

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $2$ out of $2 y$ .

$x = \frac{2 (- 3 y)}{2 (y)}$

Step 2.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 (- 3 y)}{2 y}$

Step 2.2.3.1.2.3

Rewrite the expression.

$x = \frac{- 3 y}{y}$

Step 2.2.3.2

Cancel the common factor of $y$ .

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Step 2.2.3.2.1

Cancel the common factor.

$x = \frac{- 3 y}{y}$

Step 2.2.3.2.2

Divide $- 3$ by $1$ .

$x = - 3$

Step 3

Set the denominator in $\frac{4 x + 12}{x^{2} - y^{2}}$ 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

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

Interval Notation:

$(- \infty, - 3) \cup (- 3, \infty)$

Set-Builder Notation:

${x | x \neq - 3}$