Simplify (2/x-1)/((x-7)/10+1/x)

$\frac{\frac{2}{x} - 1}{\frac{x - 7}{10} + \frac{1}{x}}$

Multiply the numerator and denominator of the complex fraction by $x \cdot 10$ .

Multiply $\frac{\frac{2}{x} - 1}{\frac{x - 7}{10} + \frac{1}{x}}$ by $\frac{x \cdot 10}{x \cdot 10}$ .

$\frac{x \cdot 10}{x \cdot 10} \cdot \frac{\frac{2}{x} - 1}{\frac{x - 7}{10} + \frac{1}{x}}$

Combine.

$\frac{x \cdot 10 (\frac{2}{x} - 1)}{x \cdot 10 (\frac{x - 7}{10} + \frac{1}{x})}$

Apply the distributive property.

$\frac{x \cdot 10 \frac{2}{x} + x \cdot 10 \cdot - 1}{x \cdot 10 \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 10$ .

$\frac{x (10) \frac{2}{x} + x \cdot 10 \cdot - 1}{x \cdot 10 \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Cancel the common factor.

$\frac{x \cdot 10 \frac{2}{x} + x \cdot 10 \cdot - 1}{x \cdot 10 \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Rewrite the expression.

$\frac{10 \cdot 2 + x \cdot 10 \cdot - 1}{x \cdot 10 \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Multiply $10$ by $2$ .

$\frac{20 + x \cdot 10 \cdot - 1}{x \cdot 10 \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Cancel the common factor of $10$ .

Factor $10$ out of $x \cdot 10$ .

$\frac{20 + x \cdot 10 \cdot - 1}{10 \cdot x \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Cancel the common factor.

$\frac{20 + x \cdot 10 \cdot - 1}{10 \cdot x \frac{x - 7}{10} + x \cdot 10 \frac{1}{x}}$

Rewrite the expression.

$\frac{20 + x \cdot 10 \cdot - 1}{x (x - 7) + x \cdot 10 \frac{1}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 10$ .

$\frac{20 + x \cdot 10 \cdot - 1}{x (x - 7) + x (10) \frac{1}{x}}$

Cancel the common factor.

$\frac{20 + x \cdot 10 \cdot - 1}{x (x - 7) + x \cdot 10 \frac{1}{x}}$

Rewrite the expression.

$\frac{20 + x \cdot 10 \cdot - 1}{x (x - 7) + 10}$

Simplify the numerator.

Move $10$ to the left of $x$ .

$\frac{20 + 10 \cdot x \cdot - 1}{x (x - 7) + 10}$

Multiply $- 1$ by $10$ .

$\frac{20 - 10 x}{x (x - 7) + 10}$

Factor $10$ out of $20 - 10 x$ .

Factor $10$ out of $20$ .

$\frac{10 (2) - 10 x}{x (x - 7) + 10}$

Factor $10$ out of $- 10 x$ .

$\frac{10 (2) + 10 (- x)}{x (x - 7) + 10}$

Factor $10$ out of $10 (2) + 10 (- x)$ .

$\frac{10 (2 - x)}{x (x - 7) + 10}$

Simplify the denominator.

Apply the distributive property.

$\frac{10 (2 - x)}{x \cdot x + x \cdot - 7 + 10}$

Multiply $x$ by $x$ .

$\frac{10 (2 - x)}{x^{2} + x \cdot - 7 + 10}$

Move $- 7$ to the left of $x$ .

$\frac{10 (2 - x)}{x^{2} - 7 \cdot x + 10}$

Factor $x^{2} - 7 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

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

Cancel the common factor of $2 - x$ and $x - 2$ .

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

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

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $10$ by $- 1$ .

$\frac{- 10}{x - 5}$

Move the negative in front of the fraction.

$- \frac{10}{x - 5}$

Simplify (2/x-1)/((x-7)/10+1/x)

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