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

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

To write $\frac{3 x - 1}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

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

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

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

Multiply $\frac{3 x - 1}{x + 2}$ by $\frac{x - 1}{x - 1}$ .

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

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

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

Reorder the factors of $(x - 1) (x + 2)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(3 x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $3$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $- 3 x$ .

$\frac{3 x^{2} - 4 x + 1 - (x - 2) (x + 2)}{(x + 2) (x - 1)}$

Apply the distributive property.

$\frac{3 x^{2} - 4 x + 1 + (- x - - 2) (x + 2)}{(x + 2) (x - 1)}$

Multiply $- 1$ by $- 2$ .

$\frac{3 x^{2} - 4 x + 1 + (- x + 2) (x + 2)}{(x + 2) (x - 1)}$

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

Apply the distributive property.

$\frac{3 x^{2} - 4 x + 1 - x (x + 2) + 2 (x + 2)}{(x + 2) (x - 1)}$

Apply the distributive property.

$\frac{3 x^{2} - 4 x + 1 - x \cdot x - x \cdot 2 + 2 (x + 2)}{(x + 2) (x - 1)}$

Apply the distributive property.

$\frac{3 x^{2} - 4 x + 1 - x \cdot x - x \cdot 2 + 2 x + 2 \cdot 2}{(x + 2) (x - 1)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{3 x^{2} - 4 x + 1 - (x \cdot x) - x \cdot 2 + 2 x + 2 \cdot 2}{(x + 2) (x - 1)}$

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 4 x + 1 - x^{2} - x \cdot 2 + 2 x + 2 \cdot 2}{(x + 2) (x - 1)}$

Multiply $2$ by $- 1$ .

$\frac{3 x^{2} - 4 x + 1 - x^{2} - 2 x + 2 x + 2 \cdot 2}{(x + 2) (x - 1)}$

Multiply $2$ by $2$ .

$\frac{3 x^{2} - 4 x + 1 - x^{2} - 2 x + 2 x + 4}{(x + 2) (x - 1)}$

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

$\frac{3 x^{2} - 4 x + 1 - x^{2} + 0 + 4}{(x + 2) (x - 1)}$

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

$\frac{3 x^{2} - 4 x + 1 - x^{2} + 4}{(x + 2) (x - 1)}$

Subtract $x^{2}$ from $3 x^{2}$ .

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

Add $1$ and $4$ .

$\frac{2 x^{2} - 4 x + 5}{(x + 2) (x - 1)}$

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

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