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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Subtract $2 x$ from $x$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $3$ .

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

Expand $(- 2 x - 3) (x + 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $1$ .

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

Multiply $- 3$ by $1$ .

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

Subtract $3 x$ from $- 2 x$ .

$\frac{2 x^{2} - x - 1 - 2 x^{2} - 5 x - 3}{(2 x + 1) (x + 1)}$

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

$\frac{- x - 1 + 0 - 5 x - 3}{(2 x + 1) (x + 1)}$

Add $- x$ and $0$ .

$\frac{- x - 1 - 5 x - 3}{(2 x + 1) (x + 1)}$

Subtract $5 x$ from $- x$ .

$\frac{- 6 x - 1 - 3}{(2 x + 1) (x + 1)}$

Subtract $3$ from $- 1$ .

$\frac{- 6 x - 4}{(2 x + 1) (x + 1)}$

Factor $2$ out of $- 6 x - 4$ .

Factor $2$ out of $- 6 x$ .

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

Factor $2$ out of $- 4$ .

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

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

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

Step 6

Simplify with factoring out.

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

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

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

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

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

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

Simplify the expression.

Rewrite $- (3 x + 2)$ as $- 1 (3 x + 2)$ .

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

Move the negative in front of the fraction.

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

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

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