Algebra Examples

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

Step 1

Add $1$ to both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Multiply $x$ by $x$ .

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

Step 2.5.3

Multiply $x$ by $1$ .

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

Step 2.5.4

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

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Step 2.5.4.1

Apply the distributive property.

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

Step 2.5.4.2

Apply the distributive property.

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

Step 2.5.4.3

Apply the distributive property.

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

Step 2.5.5

Simplify and combine like terms.

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Step 2.5.5.1

Simplify each term.

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Step 2.5.5.1.1

Multiply $x$ by $x$ .

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

Step 2.5.5.1.2

Move $- 2$ to the left of $x$ .

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

Step 2.5.5.1.3

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

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

Step 2.5.5.1.4

Multiply $- 1$ by $- 2$ .

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

Step 2.5.5.2

Subtract $x$ from $- 2 x$ .

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

Step 2.5.6

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

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

Step 2.5.7

Subtract $3 x$ from $x$ .

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

Step 2.5.8

Factor $2$ out of $2 x^{2} - 2 x + 2$ .

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Step 2.5.8.1

Factor $2$ out of $2 x^{2}$ .

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

Step 2.5.8.2

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

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

Step 2.5.8.3

Factor $2$ out of $2$ .

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

Step 2.5.8.4

Factor $2$ out of $2 (x^{2}) + 2 (- x)$ .

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

Step 2.5.8.5

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

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

Step 2.6

Write $1$ as a fraction with a common denominator.

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Simplify the numerator.

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Simplify.

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Step 2.8.2.1

Multiply $- 1$ by $2$ .

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

Step 2.8.2.2

Multiply $2$ by $1$ .

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

Step 2.8.3

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

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Step 2.8.3.1

Apply the distributive property.

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

Step 2.8.3.2

Apply the distributive property.

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

Step 2.8.3.3

Apply the distributive property.

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

Step 2.8.4

Simplify and combine like terms.

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Step 2.8.4.1

Simplify each term.

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Step 2.8.4.1.1

Multiply $x$ by $x$ .

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

Step 2.8.4.1.2

Move $- 2$ to the left of $x$ .

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

Step 2.8.4.1.3

Multiply $x$ by $1$ .

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

Step 2.8.4.1.4

Multiply $- 2$ by $1$ .

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

Step 2.8.4.2

Add $- 2 x$ and $x$ .

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

Step 2.8.5

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

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

Step 2.8.6

Subtract $x$ from $- 2 x$ .

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

Step 2.8.7

Subtract $2$ from $2$ .

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

Step 2.8.8

Add $3 x^{2} - 3 x$ and $0$ .

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

Step 2.8.9

Factor $3 x$ out of $3 x^{2} - 3 x$ .

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Step 2.8.9.1

Factor $3 x$ out of $3 x^{2}$ .

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

Step 2.8.9.2

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

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

Step 2.8.9.3

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

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

Step 3

Set the numerator equal to zero.

$3 x (x - 1) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 1 = 0$

Step 4.2

Set $x$ equal to $0$ .

$x = 0$

Step 4.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 4.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.4

The final solution is all the values that make $3 x (x - 1) = 0$ true.

$x = 0, 1$