Calculus Examples

$20 (\frac{2}{x + 1} - \frac{3}{x})$

Step 1

To write $\frac{2}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$20 (\frac{2}{x + 1} \cdot \frac{x}{x} - \frac{3}{x})$

Step 2

To write $- \frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$20 (\frac{2}{x + 1} \cdot \frac{x}{x} - \frac{3}{x} \cdot \frac{x + 1}{x + 1})$

Step 3

Write each expression with a common denominator of $(x + 1) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2}{x + 1}$ by $\frac{x}{x}$ .

$20 (\frac{2 x}{(x + 1) x} - \frac{3}{x} \cdot \frac{x + 1}{x + 1})$

Step 3.2

Multiply $\frac{3}{x}$ by $\frac{x + 1}{x + 1}$ .

$20 (\frac{2 x}{(x + 1) x} - \frac{3 (x + 1)}{x (x + 1)})$

Step 3.3

Reorder the factors of $(x + 1) x$ .

$20 (\frac{2 x}{x (x + 1)} - \frac{3 (x + 1)}{x (x + 1)})$

Step 4

Combine the numerators over the common denominator.

$20 \frac{2 x - 3 (x + 1)}{x (x + 1)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$20 \frac{2 x - 3 x - 3 \cdot 1}{x (x + 1)}$

Step 5.2

Multiply $- 3$ by $1$ .

$20 \frac{2 x - 3 x - 3}{x (x + 1)}$

Step 5.3

Subtract $3 x$ from $2 x$ .

$20 \frac{- x - 3}{x (x + 1)}$

Step 6

Simplify terms.

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Step 6.1

Combine $20$ and $\frac{- x - 3}{x (x + 1)}$ .

$\frac{20 (- x - 3)}{x (x + 1)}$

Step 6.2

Factor $- 1$ out of $- x$ .

$\frac{20 (- (x) - 3)}{x (x + 1)}$

Step 6.3

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{20 (- (x) - 1 (3))}{x (x + 1)}$

Step 6.4

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{20 (- (x + 3))}{x (x + 1)}$

Step 6.5

Simplify the expression.

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Step 6.5.1

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{20 (- 1 (x + 3))}{x (x + 1)}$

Step 6.5.2

Move the negative in front of the fraction.

$- \frac{20 (x + 3)}{x (x + 1)}$