Simplify (1/(x-3)+4/x)/(4/x-1/(x-3))

$\frac{\frac{1}{x - 3} + \frac{4}{x}}{\frac{4}{x} - \frac{1}{x - 3}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $(x - 3) x$ .

Multiply $\frac{\frac{1}{x - 3} + \frac{4}{x}}{\frac{4}{x} - \frac{1}{x - 3}}$ by $\frac{(x - 3) x}{(x - 3) x}$ .

$\frac{(x - 3) x}{(x - 3) x} \cdot \frac{\frac{1}{x - 3} + \frac{4}{x}}{\frac{4}{x} - \frac{1}{x - 3}}$

Combine.

$\frac{(x - 3) x (\frac{1}{x - 3} + \frac{4}{x})}{(x - 3) x (\frac{4}{x} - \frac{1}{x - 3})}$

Step 2

Apply the distributive property.

$\frac{(x - 3) x \frac{1}{x - 3} + (x - 3) x \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $x - 3$ .

Factor $x - 3$ out of $(x - 3) x$ .

$\frac{(x - 3) (x) \frac{1}{x - 3} + (x - 3) x \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor.

$\frac{(x - 3) x \frac{1}{x - 3} + (x - 3) x \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Rewrite the expression.

$\frac{x + (x - 3) x \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor of $x$ .

Factor $x$ out of $(x - 3) x$ .

$\frac{x + x (x - 3) \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor.

$\frac{x + x (x - 3) \frac{4}{x}}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Rewrite the expression.

$\frac{x + (x - 3) \cdot 4}{(x - 3) x \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor of $x$ .

Factor $x$ out of $(x - 3) x$ .

$\frac{x + (x - 3) \cdot 4}{x (x - 3) \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor.

$\frac{x + (x - 3) \cdot 4}{x (x - 3) \frac{4}{x} + (x - 3) x (- \frac{1}{x - 3})}$

Rewrite the expression.

$\frac{x + (x - 3) \cdot 4}{(x - 3) \cdot 4 + (x - 3) x (- \frac{1}{x - 3})}$

Cancel the common factor of $x - 3$ .

Move the leading negative in $- \frac{1}{x - 3}$ into the numerator.

$\frac{x + (x - 3) \cdot 4}{(x - 3) \cdot 4 + (x - 3) x \frac{- 1}{x - 3}}$

Factor $x - 3$ out of $(x - 3) x$ .

$\frac{x + (x - 3) \cdot 4}{(x - 3) \cdot 4 + (x - 3) (x) \frac{- 1}{x - 3}}$

Cancel the common factor.

$\frac{x + (x - 3) \cdot 4}{(x - 3) \cdot 4 + (x - 3) x \frac{- 1}{x - 3}}$

Rewrite the expression.

$\frac{x + (x - 3) \cdot 4}{(x - 3) \cdot 4 + x \cdot - 1}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{x + x \cdot 4 - 3 \cdot 4}{(x - 3) \cdot 4 + x \cdot - 1}$

Move $4$ to the left of $x$ .

$\frac{x + 4 \cdot x - 3 \cdot 4}{(x - 3) \cdot 4 + x \cdot - 1}$

Multiply $- 3$ by $4$ .

$\frac{x + 4 \cdot x - 12}{(x - 3) \cdot 4 + x \cdot - 1}$

Add $x$ and $4 x$ .

$\frac{5 x - 12}{(x - 3) \cdot 4 + x \cdot - 1}$

Step 5

Simplify the denominator.

Apply the distributive property.

$\frac{5 x - 12}{x \cdot 4 - 3 \cdot 4 + x \cdot - 1}$

Move $4$ to the left of $x$ .

$\frac{5 x - 12}{4 \cdot x - 3 \cdot 4 + x \cdot - 1}$

Multiply $- 3$ by $4$ .

$\frac{5 x - 12}{4 \cdot x - 12 + x \cdot - 1}$

Move $- 1$ to the left of $x$ .

$\frac{5 x - 12}{4 x - 12 - 1 \cdot x}$

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

$\frac{5 x - 12}{4 x - 12 - x}$

Subtract $x$ from $4 x$ .

$\frac{5 x - 12}{3 x - 12}$

Factor $3$ out of $3 x - 12$ .

Factor $3$ out of $3 x$ .

$\frac{5 x - 12}{3 (x) - 12}$

Factor $3$ out of $- 12$ .

$\frac{5 x - 12}{3 x + 3 \cdot - 4}$

Factor $3$ out of $3 x + 3 \cdot - 4$ .

$\frac{5 x - 12}{3 (x - 4)}$

Simplify (1/(x-3)+4/x)/(4/x-1/(x-3))

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