Basic Math Examples

$\frac{\frac{1}{a - x} - \frac{1}{x}}{a}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\frac{1}{a - x} \cdot \frac{x}{x} - \frac{1}{x}}{a}$

Step 1.2

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

$\frac{\frac{1}{a - x} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{a - x}{a - x}}{a}$

Step 1.3

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

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Step 1.3.1

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

$\frac{\frac{x}{(a - x) x} - \frac{1}{x} \cdot \frac{a - x}{a - x}}{a}$

Step 1.3.2

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

$\frac{\frac{x}{(a - x) x} - \frac{a - x}{x (a - x)}}{a}$

Step 1.3.3

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

$\frac{\frac{x}{x (a - x)} - \frac{a - x}{x (a - x)}}{a}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{x - (a - x)}{x (a - x)}}{a}$

Step 1.5

Rewrite $\frac{x - (a - x)}{x (a - x)}$ in a factored form.

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Step 1.5.1

Apply the distributive property.

$\frac{\frac{x - a - - x}{x (a - x)}}{a}$

Step 1.5.2

Multiply $- - x$ .

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Step 1.5.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{x - a + 1 x}{x (a - x)}}{a}$

Step 1.5.2.2

Multiply $x$ by $1$ .

$\frac{\frac{x - a + x}{x (a - x)}}{a}$

Step 1.5.3

Add $x$ and $x$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{- a + 2 x}{x (a - x)}$ by $\frac{1}{a}$ .

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

Step 4

Factor $- 1$ out of $- a$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

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

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

Step 7.2

Move the negative in front of the fraction.

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

Step 7.3

Reorder factors in $- \frac{a - 2 x}{x (a - x) a}$ .

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