Simplify ((x-7)/10+1/x)/(1/2-x/4)

$\frac{\frac{x - 7}{10} + \frac{1}{x}}{\frac{1}{2} - \frac{x}{4}}$

Multiply the numerator and denominator of the complex fraction by $20 x$ .

Multiply $\frac{\frac{x - 7}{10} + \frac{1}{x}}{\frac{1}{2} - \frac{x}{4}}$ by $\frac{20 x}{20 x}$ .

$\frac{20 x}{20 x} \cdot \frac{\frac{x - 7}{10} + \frac{1}{x}}{\frac{1}{2} - \frac{x}{4}}$

Combine.

$\frac{20 x (\frac{x - 7}{10} + \frac{1}{x})}{20 x (\frac{1}{2} - \frac{x}{4})}$

Apply the distributive property.

$\frac{20 x \frac{x - 7}{10} + 20 x \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Simplify by cancelling.

Cancel the common factor of $10$ .

Factor $10$ out of $20 x$ .

$\frac{10 (2 x) \frac{x - 7}{10} + 20 x \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Cancel the common factor.

$\frac{10 (2 x) \frac{x - 7}{10} + 20 x \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Rewrite the expression.

$\frac{2 x (x - 7) + 20 x \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Cancel the common factor of $x$ .

Factor $x$ out of $20 x$ .

$\frac{2 x (x - 7) + x \cdot 20 \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Cancel the common factor.

$\frac{2 x (x - 7) + x \cdot 20 \frac{1}{x}}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Rewrite the expression.

$\frac{2 x (x - 7) + 20}{20 x \frac{1}{2} + 20 x (- \frac{x}{4})}$

Cancel the common factor of $2$ .

Factor $2$ out of $20 x$ .

$\frac{2 x (x - 7) + 20}{2 (10 x) \frac{1}{2} + 20 x (- \frac{x}{4})}$

Cancel the common factor.

$\frac{2 x (x - 7) + 20}{2 (10 x) \frac{1}{2} + 20 x (- \frac{x}{4})}$

Rewrite the expression.

$\frac{2 x (x - 7) + 20}{10 x + 20 x (- \frac{x}{4})}$

Cancel the common factor of $4$ .

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$\frac{2 x (x - 7) + 20}{10 x + 20 x \frac{- x}{4}}$

Factor $4$ out of $20 x$ .

$\frac{2 x (x - 7) + 20}{10 x + 4 (5 x) \frac{- x}{4}}$

Cancel the common factor.

$\frac{2 x (x - 7) + 20}{10 x + 4 (5 x) \frac{- x}{4}}$

Rewrite the expression.

$\frac{2 x (x - 7) + 20}{10 x + 5 x (- x)}$

Multiply $- 1$ by $5$ .

$\frac{2 x (x - 7) + 20}{10 x - 5 x \cdot x}$

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

$\frac{2 x (x - 7) + 20}{10 x - 5 (x^{1} x)}$

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

$\frac{2 x (x - 7) + 20}{10 x - 5 (x^{1} x^{1})}$

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

$\frac{2 x (x - 7) + 20}{10 x - 5 x^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 x (x - 7) + 20}{10 x - 5 x^{2}}$

Simplify the numerator.

Factor $2$ out of $2 x (x - 7) + 20$ .

Factor $2$ out of $2 x (x - 7)$ .

$\frac{2 (x (x - 7)) + 20}{10 x - 5 x^{2}}$

Factor $2$ out of $20$ .

$\frac{2 (x (x - 7)) + 2 \cdot 10}{10 x - 5 x^{2}}$

Factor $2$ out of $2 (x (x - 7)) + 2 \cdot 10$ .

$\frac{2 (x (x - 7) + 10)}{10 x - 5 x^{2}}$

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot - 7 + 10)}{10 x - 5 x^{2}}$

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x \cdot - 7 + 10)}{10 x - 5 x^{2}}$

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

$\frac{2 (x^{2} - 7 \cdot x + 10)}{10 x - 5 x^{2}}$

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.

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

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

Simplify terms.

Factor $5 x$ out of $10 x - 5 x^{2}$ .

Factor $5 x$ out of $10 x$ .

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

Factor $5 x$ out of $- 5 x^{2}$ .

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

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

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

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

Factor $- 1$ out of $x$ .

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

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the expression.

Multiply $- 1$ by $2$ .

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

Move the negative in front of the fraction.

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

Simplify ((x-7)/10+1/x)/(1/2-x/4)

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