Precalculus Examples

$(x^{2} - 4 y^{2}) \div \frac{5 x - 10 y}{x}$

Step 1

Set the denominator in $\frac{5 x - 10 y}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 2

Set the denominator in $(x^{2} - 4 y^{2}) \div \frac{5 x - 10 y}{x}$ equal to $0$ to find where the expression is undefined.

$\frac{5 x - 10 y}{x} = 0$

Step 3

Solve for $x$ .

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Step 3.1

Set the numerator equal to zero.

$5 x - 10 y = 0$

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

Add $10 y$ to both sides of the equation.

$5 x = 10 y$

Step 3.2.2

Divide each term in $5 x = 10 y$ by $5$ and simplify.

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Step 3.2.2.1

Divide each term in $5 x = 10 y$ by $5$ .

$\frac{5 x}{5} = \frac{10 y}{5}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{10 y}{5}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10 y}{5}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Cancel the common factor of $10$ and $5$ .

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Step 3.2.2.3.1.1

Factor $5$ out of $10 y$ .

$x = \frac{5 (2 y)}{5}$

Step 3.2.2.3.1.2

Cancel the common factors.

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Step 3.2.2.3.1.2.1

Factor $5$ out of $5$ .

$x = \frac{5 (2 y)}{5 (1)}$

Step 3.2.2.3.1.2.2

Cancel the common factor.

$x = \frac{5 (2 y)}{5 \cdot 1}$

Step 3.2.2.3.1.2.3

Rewrite the expression.

$x = \frac{2 y}{1}$

Step 3.2.2.3.1.2.4

Divide $2 y$ by $1$ .

$x = 2 y$

Step 4

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

Interval Notation:

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

Set-Builder Notation:

${x | x \neq 0}$