Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

$\frac{x}{x + 1} + \frac{x}{x - 2} - 2 = 0$

Step 2

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

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Step 2.1

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

$\frac{x}{x + 1} \cdot \frac{x - 2}{x - 2} + \frac{x}{x - 2} - 2 = 0$

Step 2.2

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

$\frac{x}{x + 1} \cdot \frac{x - 2}{x - 2} + \frac{x}{x - 2} \cdot \frac{x + 1}{x + 1} - 2 = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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

Step 2.5.2

Add $x$ and $x$ .

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

Step 2.5.3

Add $- 2$ and $1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

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

Step 2.9.2

Rewrite using the commutative property of multiplication.

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

Step 2.9.3

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

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

Step 2.9.4

Simplify each term.

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Step 2.9.4.1

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

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Step 2.9.4.1.1

Move $x$ .

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

Step 2.9.4.1.2

Multiply $x$ by $x$ .

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

Step 2.9.4.2

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

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

Step 2.9.5

Apply the distributive property.

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

Step 2.9.6

Multiply $- 2$ by $1$ .

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

Step 2.9.7

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

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Step 2.9.7.1

Apply the distributive property.

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

Step 2.9.7.2

Apply the distributive property.

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

Step 2.9.7.3

Apply the distributive property.

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

Step 2.9.8

Simplify and combine like terms.

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Step 2.9.8.1

Simplify each term.

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Step 2.9.8.1.1

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

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Step 2.9.8.1.1.1

Move $x$ .

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

Step 2.9.8.1.1.2

Multiply $x$ by $x$ .

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

Step 2.9.8.1.2

Multiply $- 2$ by $- 2$ .

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

Step 2.9.8.1.3

Multiply $- 2$ by $- 2$ .

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

Step 2.9.8.2

Subtract $2 x$ from $4 x$ .

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

Step 2.9.9

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

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

Step 2.9.10

Add $- x$ and $0$ .

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

Step 2.9.11

Add $- x$ and $2 x$ .

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

Step 3

Set the numerator equal to zero.

$x + 4 = 0$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$