Precalculus Examples

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

Step 1

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

$x + y = 0$

Step 2

Subtract $y$ from both sides of the equation.

$x = - y$

Step 3

Set the denominator in $\frac{3 x + 3 y}{x^{2} - y^{2}}$ 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 real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$