Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move $x$ .

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

Step 5.2.1.1.2

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Multiply $2$ by $1$ .

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

Step 5.2.1.3

Multiply $- 3$ by $1$ .

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

Step 5.2.2

Subtract $3 x$ from $2 x$ .

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Multiply $2$ by $- 1$ .

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

Step 5.5

Multiply $- 1$ by $1$ .

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

Step 5.6

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

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Step 5.6.1

Apply the distributive property.

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

Step 5.6.2

Apply the distributive property.

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

Step 5.6.3

Apply the distributive property.

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

Step 5.7

Simplify and combine like terms.

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Step 5.7.1

Simplify each term.

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Step 5.7.1.1

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

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Step 5.7.1.1.1

Move $x$ .

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

Step 5.7.1.1.2

Multiply $x$ by $x$ .

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

Step 5.7.1.2

Multiply $- 1$ by $- 2$ .

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

Step 5.7.1.3

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

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

Step 5.7.1.4

Multiply $- 1$ by $- 1$ .

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

Step 5.7.2

Subtract $x$ from $2 x$ .

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

Step 5.8

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

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

Step 5.9

Add $- x$ and $0$ .

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

Step 5.10

Add $- x$ and $x$ .

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

Step 5.11

Subtract $3$ from $0$ .

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

Step 5.12

Add $- 3$ and $1$ .

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

Step 6

Move the negative in front of the fraction.

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