Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

Tap for more steps...

Step 5.2.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 5.2.1.3

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

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

Step 5.2.1.4

Multiply $x$ by $1$ .

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

Step 5.2.1.5

Multiply $- 1$ by $1$ .

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

Step 5.2.2

Add $- x$ and $x$ .

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

Step 5.2.3

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Apply the distributive property.

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

Step 5.3.3

Apply the distributive property.

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

Step 5.4

Combine the opposite terms in $x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2$ .

Tap for more steps...

Step 5.4.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

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

Step 5.4.2

Subtract $2 x$ from $2 x$ .

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

Step 5.4.3

Add $x \cdot x$ and $0$ .

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

Step 5.5

Simplify each term.

Tap for more steps...

Step 5.5.1

Multiply $x$ by $x$ .

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

Step 5.5.2

Multiply $- 2$ by $2$ .

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

Step 5.6

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

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

Step 5.7

Subtract $4$ from $- 1$ .

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