Precalculus Examples

$\frac{4 y}{3 x^{2} + 2 x y} - \frac{9 x}{3 x y + 2 y^{2}}$

Step 1

Set the denominator in $\frac{4 y}{3 x^{2} + 2 x y}$ equal to $0$ to find where the expression is undefined.

$3 x^{2} + 2 x y = 0$

Step 2

Solve for $y$ .

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Step 2.1

Subtract $3 x^{2}$ from both sides of the equation.

$2 x y = - 3 x^{2}$

Step 2.2

Divide each term in $2 x y = - 3 x^{2}$ by $2 x$ and simplify.

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Step 2.2.1

Divide each term in $2 x y = - 3 x^{2}$ by $2 x$ .

$\frac{2 x y}{2 x} = \frac{- 3 x^{2}}{2 x}$

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.

$\frac{2 x y}{2 x} = \frac{- 3 x^{2}}{2 x}$

Step 2.2.2.1.2

Rewrite the expression.

$\frac{x y}{x} = \frac{- 3 x^{2}}{2 x}$

Step 2.2.2.2

Cancel the common factor of $x$ .

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Step 2.2.2.2.1

Cancel the common factor.

$\frac{x y}{x} = \frac{- 3 x^{2}}{2 x}$

Step 2.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 3 x^{2}}{2 x}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 2.2.3.1.1

Factor $x$ out of $- 3 x^{2}$ .

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

Step 2.2.3.1.2

Cancel the common factors.

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Step 2.2.3.1.2.1

Factor $x$ out of $2 x$ .

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

Step 2.2.3.1.2.2

Cancel the common factor.

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

Step 2.2.3.1.2.3

Rewrite the expression.

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

Step 2.2.3.2

Move the negative in front of the fraction.

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

Step 3

Set the denominator in $\frac{9 x}{3 x y + 2 y^{2}}$ equal to $0$ to find where the expression is undefined.

$3 x y + 2 y^{2} = 0$

Step 4

Solve for $y$ .

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Step 4.1

Factor $y$ out of $3 x y + 2 y^{2}$ .

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Step 4.1.1

Factor $y$ out of $3 x y$ .

$y (3 x) + 2 y^{2} = 0$

Step 4.1.2

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

$y (3 x) + y (2 y) = 0$

Step 4.1.3

Factor $y$ out of $y (3 x) + y (2 y)$ .

$y (3 x + 2 y) = 0$

Step 4.2

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

$y = 0$

$3 x + 2 y = 0$

Step 4.3

Set $y$ equal to $0$ .

$y = 0$

Step 4.4

Set $3 x + 2 y$ equal to $0$ and solve for $y$ .

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Step 4.4.1

Set $3 x + 2 y$ equal to $0$ .

$3 x + 2 y = 0$

Step 4.4.2

Solve $3 x + 2 y = 0$ for $y$ .

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Step 4.4.2.1

Subtract $3 x$ from both sides of the equation.

$2 y = - 3 x$

Step 4.4.2.2

Divide each term in $2 y = - 3 x$ by $2$ and simplify.

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Step 4.4.2.2.1

Divide each term in $2 y = - 3 x$ by $2$ .

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

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

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

Step 4.4.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 4.4.2.2.3

Simplify the right side.

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Step 4.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 4.5

The final solution is all the values that make $y (3 x + 2 y) = 0$ true.

$y = 0$

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

$y = 0$

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

Step 5

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

Interval Notation:

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

Set-Builder Notation:

${y | y \neq 0}$