Algebra Examples

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

Step 1

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

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

Step 2

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

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

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 $\frac{2}{x}$ by $\frac{x - 5}{x - 5}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $2$ by $- 5$ .

$\frac{2 x - 10 + (x - 1) x}{x (x - 5)}$

Step 5.3

Apply the distributive property.

$\frac{2 x - 10 + x \cdot x - 1 x}{x (x - 5)}$

Step 5.4

Multiply $x$ by $x$ .

$\frac{2 x - 10 + x^{2} - 1 x}{x (x - 5)}$

Step 5.5

Rewrite $- 1 x$ as $- x$ .

$\frac{2 x - 10 + x^{2} - x}{x (x - 5)}$

Step 5.6

Subtract $x$ from $2 x$ .

$\frac{x - 10 + x^{2}}{x (x - 5)}$

Step 5.7

Reorder terms.

$\frac{x^{2} + x - 10}{x (x - 5)}$