Simplify -6/(x-1)+(3-3x)/x

$- \frac{6}{x - 1} + \frac{3 - 3 x}{x}$

Step 1

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

Factor $3$ out of $3$ .

$- \frac{6}{x - 1} + \frac{3 (1) - 3 x}{x}$

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

$- \frac{6}{x - 1} + \frac{3 (1) + 3 (- x)}{x}$

Factor $3$ out of $3 (1) + 3 (- x)$ .

$- \frac{6}{x - 1} + \frac{3 (1 - x)}{x}$

Step 2

To write $- \frac{6}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{6}{x - 1} \cdot \frac{x}{x} + \frac{3 (1 - x)}{x}$

Step 3

To write $\frac{3 (1 - x)}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$- \frac{6}{x - 1} \cdot \frac{x}{x} + \frac{3 (1 - x)}{x} \cdot \frac{x - 1}{x - 1}$

Step 4

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

Multiply $\frac{6}{x - 1}$ by $\frac{x}{x}$ .

$- \frac{6 x}{(x - 1) x} + \frac{3 (1 - x)}{x} \cdot \frac{x - 1}{x - 1}$

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

$- \frac{6 x}{(x - 1) x} + \frac{3 (1 - x) (x - 1)}{x (x - 1)}$

Reorder the factors of $(x - 1) x$ .

$- \frac{6 x}{x (x - 1)} + \frac{3 (1 - x) (x - 1)}{x (x - 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{- 6 x + 3 (1 - x) (x - 1)}{x (x - 1)}$

Step 6

Simplify the numerator.

Factor $3$ out of $- 6 x + 3 (1 - x) (x - 1)$ .

Factor $3$ out of $- 6 x$ .

$\frac{3 (- 2 x) + 3 (1 - x) (x - 1)}{x (x - 1)}$

Factor $3$ out of $3 (1 - x) (x - 1)$ .

$\frac{3 (- 2 x) + 3 ((1 - x) (x - 1))}{x (x - 1)}$

Factor $3$ out of $3 (- 2 x) + 3 ((1 - x) (x - 1))$ .

$\frac{3 (- 2 x + (1 - x) (x - 1))}{x (x - 1)}$

Expand $(1 - x) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{3 (- 2 x + 1 (x - 1) - x (x - 1))}{x (x - 1)}$

Apply the distributive property.

$\frac{3 (- 2 x + 1 x + 1 \cdot - 1 - x (x - 1))}{x (x - 1)}$

Apply the distributive property.

$\frac{3 (- 2 x + 1 x + 1 \cdot - 1 - x \cdot x - x \cdot - 1)}{x (x - 1)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $1$ .

$\frac{3 (- 2 x + x + 1 \cdot - 1 - x \cdot x - x \cdot - 1)}{x (x - 1)}$

Multiply $- 1$ by $1$ .

$\frac{3 (- 2 x + x - 1 - x \cdot x - x \cdot - 1)}{x (x - 1)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{3 (- 2 x + x - 1 - (x \cdot x) - x \cdot - 1)}{x (x - 1)}$

Multiply $x$ by $x$ .

$\frac{3 (- 2 x + x - 1 - x^{2} - x \cdot - 1)}{x (x - 1)}$

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$\frac{3 (- 2 x + x - 1 - x^{2} + 1 x)}{x (x - 1)}$

Multiply $x$ by $1$ .

$\frac{3 (- 2 x + x - 1 - x^{2} + x)}{x (x - 1)}$

Add $x$ and $x$ .

$\frac{3 (- 2 x + 2 x - 1 - x^{2})}{x (x - 1)}$

Add $- 2 x$ and $2 x$ .

$\frac{3 (0 - 1 - x^{2})}{x (x - 1)}$

Subtract $1$ from $0$ .

$\frac{3 (- 1 - x^{2})}{x (x - 1)}$

Step 7

Simplify with factoring out.

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

$\frac{3 (- 1 (1) - x^{2})}{x (x - 1)}$

Factor $- 1$ out of $- x^{2}$ .

$\frac{3 (- 1 (1) - (x^{2}))}{x (x - 1)}$

Factor $- 1$ out of $- 1 (1) - (x^{2})$ .

$\frac{3 (- 1 (1 + x^{2}))}{x (x - 1)}$

Move the negative in front of the fraction.

$- \frac{3 (1 + x^{2})}{x (x - 1)}$

Simplify -6/(x-1)+(3-3x)/x

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