Precalculus Examples

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

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

Step 1

To write

\frac{2}{y + 3}

$\frac{2}{y + 3}$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 2

To write

- \frac{1}{3}

$- \frac{1}{3}$ as a fraction with a common denominator, multiply by

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

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

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

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

Step 3

Write each expression with a common denominator of

(y + 3) \cdot 3

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

1

$1$ .

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Step 3.1

Multiply

\frac{2}{y + 3}

$\frac{2}{y + 3}$ by

\frac{3}{3}

$\frac{3}{3}$ .

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

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

Step 3.2

Multiply

\frac{1}{3}

$\frac{1}{3}$ by

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

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

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

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

Step 3.3

Reorder the factors of

(y + 3) \cdot 3

$(y + 3) \cdot 3$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Multiply

2

$2$ by

3

$3$ .

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

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

Step 5.2

Apply the distributive property.

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

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

Step 5.3

Multiply

- 1

$- 1$ by

3

$3$ .

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

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

Step 5.4

Subtract

3

$3$ from

6

$6$ .

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

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

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor

- 1

$- 1$ out of

- y

$- y$ .

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

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

Step 6.2

Rewrite

3

$3$ as

- 1 (- 3)

$- 1 (- 3)$ .

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

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

Step 6.3

Factor

- 1

$- 1$ out of

- (y) - 1 (- 3)

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

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

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite

- (y - 3)

$- (y - 3)$ as

- 1 (y - 3)

$- 1 (y - 3)$ .

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

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

Step 6.4.2

Move the negative in front of the fraction.

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

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

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

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

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

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

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