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

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $x (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 3) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

Subtract $x$ from $x$ .

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

Add $- 3 + x^{2}$ and $0$ .

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

Step 6

Simplify with factoring out.

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

Factor $- 1$ out of $- 1 (3) - 1 (- x^{2})$ .

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

Move the negative in front of the fraction.

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