Calculus Examples

\frac{- 3}{y + 3} + \frac{1}{- 2 y}

$\frac{- 3}{y + 3} + \frac{1}{- 2 y}$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

- \frac{3}{y + 3} + \frac{1}{- 2 y}

$- \frac{3}{y + 3} + \frac{1}{- 2 y}$

Step 1.2

Move the negative in front of the fraction.

- \frac{3}{y + 3} - \frac{1}{2 y}

$- \frac{3}{y + 3} - \frac{1}{2 y}$

- \frac{3}{y + 3} - \frac{1}{2 y}

$- \frac{3}{y + 3} - \frac{1}{2 y}$

Step 2

To write

- \frac{3}{y + 3}

$- \frac{3}{y + 3}$ as a fraction with a common denominator, multiply by

\frac{2 y}{2 y}

$\frac{2 y}{2 y}$ .

- \frac{3}{y + 3} \cdot \frac{2 y}{2 y} - \frac{1}{2 y}

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

Step 3

To write

- \frac{1}{2 y}

$- \frac{1}{2 y}$ as a fraction with a common denominator, multiply by

\frac{y + 3}{y + 3}

$\frac{y + 3}{y + 3}$ .

- \frac{3}{y + 3} \cdot \frac{2 y}{2 y} - \frac{1}{2 y} \cdot \frac{y + 3}{y + 3}

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

Step 4

Write each expression with a common denominator of

(y + 3) \cdot 2 y

$(y + 3) \cdot 2 y$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 4.1

Multiply

\frac{3}{y + 3}

$\frac{3}{y + 3}$ by

\frac{2 y}{2 y}

$\frac{2 y}{2 y}$ .

- \frac{3 (2 y)}{(y + 3) (2 y)} - \frac{1}{2 y} \cdot \frac{y + 3}{y + 3}

$- \frac{3 (2 y)}{(y + 3) (2 y)} - \frac{1}{2 y} \cdot \frac{y + 3}{y + 3}$

Step 4.2

Multiply

\frac{1}{2 y}

$\frac{1}{2 y}$ by

\frac{y + 3}{y + 3}

$\frac{y + 3}{y + 3}$ .

- \frac{3 (2 y)}{(y + 3) (2 y)} - \frac{y + 3}{2 y (y + 3)}

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

Step 4.3

Reorder the factors of

(y + 3) (2 y)

$(y + 3) (2 y)$ .

- \frac{3 (2 y)}{2 y (y + 3)} - \frac{y + 3}{2 y (y + 3)}

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

- \frac{3 (2 y)}{2 y (y + 3)} - \frac{y + 3}{2 y (y + 3)}

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

Step 5

Combine the numerators over the common denominator.

\frac{- 3 (2 y) - (y + 3)}{2 y (y + 3)}

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply

- 3

$- 3$ by

2

$2$ .

\frac{- 6 y - (y + 3)}{2 y (y + 3)}

$\frac{- 6 y - (y + 3)}{2 y (y + 3)}$

Step 6.2

Apply the distributive property.

\frac{- 6 y - y - 1 \cdot 3}{2 y (y + 3)}

$\frac{- 6 y - y - 1 \cdot 3}{2 y (y + 3)}$

Step 6.3

Multiply

- 1

$- 1$ by

3

$3$ .

\frac{- 6 y - y - 3}{2 y (y + 3)}

$\frac{- 6 y - y - 3}{2 y (y + 3)}$

Step 6.4

Subtract

y

$y$ from

- 6 y

$- 6 y$ .

\frac{- 7 y - 3}{2 y (y + 3)}

$\frac{- 7 y - 3}{2 y (y + 3)}$

\frac{- 7 y - 3}{2 y (y + 3)}

$\frac{- 7 y - 3}{2 y (y + 3)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor

- 1

$- 1$ out of

- 7 y

$- 7 y$ .

\frac{- (7 y) - 3}{2 y (y + 3)}

$\frac{- (7 y) - 3}{2 y (y + 3)}$

Step 7.2

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

\frac{- (7 y) - 1 (3)}{2 y (y + 3)}

$\frac{- (7 y) - 1 (3)}{2 y (y + 3)}$

Step 7.3

Factor

- 1

$- 1$ out of

- (7 y) - 1 (3)

$- (7 y) - 1 (3)$ .

\frac{- (7 y + 3)}{2 y (y + 3)}

$\frac{- (7 y + 3)}{2 y (y + 3)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite

- (7 y + 3)

$- (7 y + 3)$ as

- 1 (7 y + 3)

$- 1 (7 y + 3)$ .

\frac{- 1 (7 y + 3)}{2 y (y + 3)}

$\frac{- 1 (7 y + 3)}{2 y (y + 3)}$

Step 7.4.2

Move the negative in front of the fraction.

- \frac{7 y + 3}{2 y (y + 3)}

$- \frac{7 y + 3}{2 y (y + 3)}$

- \frac{7 y + 3}{2 y (y + 3)}

$- \frac{7 y + 3}{2 y (y + 3)}$

- \frac{7 y + 3}{2 y (y + 3)}

$- \frac{7 y + 3}{2 y (y + 3)}$

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