Precalculus Examples

$(\frac{3}{x}) - (\frac{5}{x + 5})$

Step 1

To write $\frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3}{x} \cdot \frac{x + 5}{x + 5} - \frac{5}{x + 5}$

Step 2

To write $- \frac{5}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3}{x} \cdot \frac{x + 5}{x + 5} - \frac{5}{x + 5} \cdot \frac{x}{x}$

Step 3

Write each expression with a common denominator of $x (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3}{x}$ by $\frac{x + 5}{x + 5}$ .

$\frac{3 (x + 5)}{x (x + 5)} - \frac{5}{x + 5} \cdot \frac{x}{x}$

Step 3.2

Multiply $\frac{5}{x + 5}$ by $\frac{x}{x}$ .

$\frac{3 (x + 5)}{x (x + 5)} - \frac{5 x}{(x + 5) x}$

Step 3.3

Reorder the factors of $(x + 5) x$ .

$\frac{3 (x + 5)}{x (x + 5)} - \frac{5 x}{x (x + 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (x + 5) - 5 x}{x (x + 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 5 - 5 x}{x (x + 5)}$

Step 5.2

Multiply $3$ by $5$ .

$\frac{3 x + 15 - 5 x}{x (x + 5)}$

Step 5.3

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x + 15}{x (x + 5)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) + 15}{x (x + 5)}$

Step 6.2

Rewrite $15$ as $- 1 (- 15)$ .

$\frac{- (2 x) - 1 (- 15)}{x (x + 5)}$

Step 6.3

Factor $- 1$ out of $- (2 x) - 1 (- 15)$ .

$\frac{- (2 x - 15)}{x (x + 5)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (2 x - 15)$ as $- 1 (2 x - 15)$ .

$\frac{- 1 (2 x - 15)}{x (x + 5)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{2 x - 15}{x (x + 5)}$