Algebra Examples

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

Step 1

To write $- \frac{3 x + 1}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2

Write each expression with a common denominator of $2 (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{3 x + 1}{x - 5}$ by $\frac{2}{2}$ .

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

Step 2.2

Reorder the factors of $(x - 5) \cdot 2$ .

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Multiply $3$ by $- 1$ .

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

Step 4.3

Multiply $- 1$ by $1$ .

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

Step 4.4

Apply the distributive property.

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

Step 4.5

Multiply $2$ by $- 3$ .

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

Step 4.6

Multiply $- 1$ by $2$ .

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

Step 4.7

Subtract $6 x$ from $x$ .

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

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 5 x$ .

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

Step 5.2

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

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

Step 5.3

Factor $- 1$ out of $- (5 x) - 1 (2)$ .

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

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (5 x + 2)$ as $- 1 (5 x + 2)$ .

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

Step 5.4.2

Move the negative in front of the fraction.

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