Simplify -3/(x-2)+(1-x)/x

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

Step 1

To write $- \frac{3}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2

To write $\frac{1 - x}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Write each expression with a common denominator of $(x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{x - 2}$ by $\frac{x}{x}$ .

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

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

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

Reorder the factors of $(x - 2) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Expand $(1 - x) (x - 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $1$ .

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

Multiply $- 2$ by $1$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 1$ .

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

Add $x$ and $2 x$ .

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

Add $- 3 x$ and $3 x$ .

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

Subtract $2$ from $0$ .

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

Step 6

Simplify with factoring out.

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Simplify -3/(x-2)+(1-x)/x

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