Calculus Examples

$r (w) = \frac{3 w - 1}{w + 2} - 3$

Step 1

Divide each term in $r w = \frac{3 w - 1}{w + 2} - 3$ by $w$ and simplify.

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Step 1.1

Divide each term in $r w = \frac{3 w - 1}{w + 2} - 3$ by $w$ .

$\frac{r w}{w} = \frac{\frac{3 w - 1}{w + 2}}{w} + \frac{- 3}{w}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $w$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{r w}{w} = \frac{\frac{3 w - 1}{w + 2}}{w} + \frac{- 3}{w}$

Step 1.2.1.2

Divide $r$ by $1$ .

$r = \frac{\frac{3 w - 1}{w + 2}}{w} + \frac{- 3}{w}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Combine the numerators over the common denominator.

$r = \frac{\frac{3 w - 1}{w + 2} - 3}{w}$

Step 1.3.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{w + 2}{w + 2}$ .

$r = \frac{\frac{3 w - 1}{w + 2} - 3 \cdot \frac{w + 2}{w + 2}}{w}$

Step 1.3.3

Simplify terms.

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Step 1.3.3.1

Combine $- 3$ and $\frac{w + 2}{w + 2}$ .

$r = \frac{\frac{3 w - 1}{w + 2} + \frac{- 3 (w + 2)}{w + 2}}{w}$

Step 1.3.3.2

Combine the numerators over the common denominator.

$r = \frac{\frac{3 w - 1 - 3 (w + 2)}{w + 2}}{w}$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Apply the distributive property.

$r = \frac{\frac{3 w - 1 - 3 w - 3 \cdot 2}{w + 2}}{w}$

Step 1.3.4.2

Multiply $- 3$ by $2$ .

$r = \frac{\frac{3 w - 1 - 3 w - 6}{w + 2}}{w}$

Step 1.3.4.3

Subtract $3 w$ from $3 w$ .

$r = \frac{\frac{0 - 1 - 6}{w + 2}}{w}$

Step 1.3.4.4

Subtract $1$ from $0$ .

$r = \frac{\frac{- 1 - 6}{w + 2}}{w}$

Step 1.3.4.5

Subtract $6$ from $- 1$ .

$r = \frac{\frac{- 7}{w + 2}}{w}$

Step 1.3.5

Move the negative in front of the fraction.

$r = \frac{- \frac{7}{w + 2}}{w}$

Step 1.3.6

Multiply the numerator by the reciprocal of the denominator.

$r = - \frac{7}{w + 2} \cdot \frac{1}{w}$

Step 1.3.7

Multiply $\frac{1}{w}$ by $\frac{7}{w + 2}$ .

$r = - \frac{7}{w (w + 2)}$

Step 2

To rewrite as a function of $w$ , write the equation so that $r$ is by itself on one side of the equal sign and an expression involving only $w$ is on the other side.

$f (w) = - \frac{7}{w (w + 2)}$