Calculus Examples

$\frac{5}{2 x} - \frac{5}{3 x + 15}$

Step 1

Factor $3$ out of $3 x + 15$ .

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Step 1.1

Factor $3$ out of $3 x$ .

$\frac{5}{2 x} - \frac{5}{3 (x) + 15}$

Step 1.2

Factor $3$ out of $15$ .

$\frac{5}{2 x} - \frac{5}{3 x + 3 \cdot 5}$

Step 1.3

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{5}{2 x} - \frac{5}{3 (x + 5)}$

Step 2

To write $\frac{5}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3 (x + 5)}{3 (x + 5)}$ .

$\frac{5}{2 x} \cdot \frac{3 (x + 5)}{3 (x + 5)} - \frac{5}{3 (x + 5)}$

Step 3

To write $- \frac{5}{3 (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{2 x}{2 x}$ .

$\frac{5}{2 x} \cdot \frac{3 (x + 5)}{3 (x + 5)} - \frac{5}{3 (x + 5)} \cdot \frac{2 x}{2 x}$

Step 4

Write each expression with a common denominator of $6 x (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5}{2 x}$ by $\frac{3 (x + 5)}{3 (x + 5)}$ .

$\frac{5 (3 (x + 5))}{2 x (3 (x + 5))} - \frac{5}{3 (x + 5)} \cdot \frac{2 x}{2 x}$

Step 4.2

Multiply $3$ by $2$ .

$\frac{5 (3 (x + 5))}{6 x (x + 5)} - \frac{5}{3 (x + 5)} \cdot \frac{2 x}{2 x}$

Step 4.3

Multiply $\frac{5}{3 (x + 5)}$ by $\frac{2 x}{2 x}$ .

$\frac{5 (3 (x + 5))}{6 x (x + 5)} - \frac{5 (2 x)}{3 (x + 5) (2 x)}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{5 (3 (x + 5))}{6 x (x + 5)} - \frac{5 (2 x)}{6 (x + 5) x}$

Step 4.5

Reorder the factors of $6 (x + 5) x$ .

$\frac{5 (3 (x + 5))}{6 x (x + 5)} - \frac{5 (2 x)}{6 x (x + 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (3 (x + 5)) - 5 (2 x)}{6 x (x + 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $5$ out of $5 \cdot 3 (x + 5) - 5 \cdot 2 x$ .

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Step 6.1.1

Factor $5$ out of $5 \cdot 3 (x + 5)$ .

$\frac{5 (3 (x + 5)) - 5 \cdot 2 x}{6 x (x + 5)}$

Step 6.1.2

Factor $5$ out of $- 5 \cdot 2 x$ .

$\frac{5 (3 (x + 5)) + 5 (- 1 \cdot 2 x)}{6 x (x + 5)}$

Step 6.1.3

Factor $5$ out of $5 (3 (x + 5)) + 5 (- 1 \cdot 2 x)$ .

$\frac{5 (3 (x + 5) - 1 \cdot 2 x)}{6 x (x + 5)}$

Step 6.2

Apply the distributive property.

$\frac{5 (3 x + 3 \cdot 5 - 1 \cdot 2 x)}{6 x (x + 5)}$

Step 6.3

Multiply $3$ by $5$ .

$\frac{5 (3 x + 15 - 1 \cdot 2 x)}{6 x (x + 5)}$

Step 6.4

Multiply $- 1$ by $2$ .

$\frac{5 (3 x + 15 - 2 x)}{6 x (x + 5)}$

Step 6.5

Subtract $2 x$ from $3 x$ .

$\frac{5 (x + 15)}{6 x (x + 5)}$