Simplify (x/4-1/x)/(1-2/x)

$\frac{\frac{x}{4} - \frac{1}{x}}{1 - \frac{2}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $4 x$ .

Multiply $\frac{\frac{x}{4} - \frac{1}{x}}{1 - \frac{2}{x}}$ by $\frac{4 x}{4 x}$ .

$\frac{4 x}{4 x} \cdot \frac{\frac{x}{4} - \frac{1}{x}}{1 - \frac{2}{x}}$

Combine.

$\frac{4 x (\frac{x}{4} - \frac{1}{x})}{4 x (1 - \frac{2}{x})}$

Step 2

Apply the distributive property.

$\frac{4 x \frac{x}{4} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $4$ .

Factor $4$ out of $4 x$ .

$\frac{4 (x) \frac{x}{4} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Cancel the common factor.

$\frac{4 x \frac{x}{4} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Rewrite the expression.

$\frac{x \cdot x + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Raise $x$ to the power of $1$ .

$\frac{x^{1} x + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Add $1$ and $1$ .

$\frac{x^{2} + 4 x (- \frac{1}{x})}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{x^{2} + 4 x \frac{- 1}{x}}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Factor $x$ out of $4 x$ .

$\frac{x^{2} + x \cdot 4 \frac{- 1}{x}}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Cancel the common factor.

$\frac{x^{2} + x \cdot 4 \frac{- 1}{x}}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Rewrite the expression.

$\frac{x^{2} + 4 \cdot - 1}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Multiply $4$ by $- 1$ .

$\frac{x^{2} - 4}{4 x \cdot 1 + 4 x (- \frac{2}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{2}{x}$ into the numerator.

$\frac{x^{2} - 4}{4 x \cdot 1 + 4 x \frac{- 2}{x}}$

Factor $x$ out of $4 x$ .

$\frac{x^{2} - 4}{4 x \cdot 1 + x \cdot 4 \frac{- 2}{x}}$

Cancel the common factor.

$\frac{x^{2} - 4}{4 x \cdot 1 + x \cdot 4 \frac{- 2}{x}}$

Rewrite the expression.

$\frac{x^{2} - 4}{4 x \cdot 1 + 4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$\frac{x^{2} - 4}{4 x \cdot 1 - 8}$

Step 4

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{4 x \cdot 1 - 8}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{(x + 2) (x - 2)}{4 x \cdot 1 - 8}$

Step 5

Simplify the denominator.

Factor $4$ out of $4 x \cdot 1 - 8$ .

Factor $4$ out of $4 x \cdot 1$ .

$\frac{(x + 2) (x - 2)}{4 (x \cdot 1) - 8}$

Factor $4$ out of $- 8$ .

$\frac{(x + 2) (x - 2)}{4 (x \cdot 1) + 4 \cdot - 2}$

Factor $4$ out of $4 (x \cdot 1) + 4 \cdot - 2$ .

$\frac{(x + 2) (x - 2)}{4 (x \cdot 1 - 2)}$

Multiply $x$ by $1$ .

$\frac{(x + 2) (x - 2)}{4 (x - 2)}$

Step 6

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$\frac{(x + 2) (x - 2)}{4 (x - 2)}$

Rewrite the expression.

$\frac{x + 2}{4}$

Simplify (x/4-1/x)/(1-2/x)

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