Algebra Examples

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

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{4}{x - 2}$ from both sides of the equation.

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

Step 1.2

Subtract $\frac{2}{x + 1}$ from both sides of the equation.

$\frac{6}{x - 1} - \frac{4}{x - 2} - \frac{2}{x + 1} = 0$

Step 2

Simplify $\frac{6}{x - 1} - \frac{4}{x - 2} - \frac{2}{x + 1}$ .

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Step 2.1

To write $\frac{6}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{6}{x - 1} \cdot \frac{x - 2}{x - 2} - \frac{4}{x - 2} - \frac{2}{x + 1} = 0$

Step 2.2

To write $- \frac{4}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{6}{x - 1} \cdot \frac{x - 2}{x - 2} - \frac{4}{x - 2} \cdot \frac{x - 1}{x - 1} - \frac{2}{x + 1} = 0$

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 $\frac{6}{x - 1}$ by $\frac{x - 2}{x - 2}$ .

$\frac{6 (x - 2)}{(x - 1) (x - 2)} - \frac{4}{x - 2} \cdot \frac{x - 1}{x - 1} - \frac{2}{x + 1} = 0$

Step 2.3.2

Multiply $\frac{4}{x - 2}$ by $\frac{x - 1}{x - 1}$ .

$\frac{6 (x - 2)}{(x - 1) (x - 2)} - \frac{4 (x - 1)}{(x - 2) (x - 1)} - \frac{2}{x + 1} = 0$

Step 2.3.3

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

$\frac{6 (x - 2)}{(x - 1) (x - 2)} - \frac{4 (x - 1)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{6 (x - 2) - 4 (x - 1)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $2$ out of $6 (x - 2) - 4 (x - 1)$ .

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Step 2.5.1.1

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

$\frac{2 (3 (x - 2)) - 4 (x - 1)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5.1.2

Factor $2$ out of $- 4 (x - 1)$ .

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

Step 2.5.1.3

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

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $3$ by $- 2$ .

$\frac{2 (3 x - 6 - 2 (x - 1))}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5.4

Apply the distributive property.

$\frac{2 (3 x - 6 - 2 x - 2 \cdot - 1)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5.5

Multiply $- 2$ by $- 1$ .

$\frac{2 (3 x - 6 - 2 x + 2)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5.6

Subtract $2 x$ from $3 x$ .

$\frac{2 (x - 6 + 2)}{(x - 1) (x - 2)} - \frac{2}{x + 1} = 0$

Step 2.5.7

Add $- 6$ and $2$ .

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

Step 2.6

To write $\frac{2 (x - 4)}{(x - 1) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 2.7

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

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

Step 2.8

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

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Step 2.8.1

Multiply $\frac{2 (x - 4)}{(x - 1) (x - 2)}$ by $\frac{x + 1}{x + 1}$ .

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

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.8.4

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

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Simplify the numerator.

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Step 2.10.1

Factor $2$ out of $2 (x - 4) (x + 1) - 2 (x - 1) (x - 2)$ .

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Step 2.10.1.1

Factor $2$ out of $2 (x - 4) (x + 1)$ .

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

Step 2.10.1.2

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

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

Step 2.10.1.3

Factor $2$ out of $2 ((x - 4) (x + 1)) + 2 (- (x - 1) (x - 2))$ .

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

Step 2.10.2

Expand $(x - 4) (x + 1)$ using the FOIL Method.

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Step 2.10.2.1

Apply the distributive property.

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

Step 2.10.2.2

Apply the distributive property.

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

Step 2.10.2.3

Apply the distributive property.

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

Step 2.10.3

Simplify and combine like terms.

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Step 2.10.3.1

Simplify each term.

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Step 2.10.3.1.1

Multiply $x$ by $x$ .

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

Step 2.10.3.1.2

Multiply $x$ by $1$ .

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

Step 2.10.3.1.3

Multiply $- 4$ by $1$ .

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

Step 2.10.3.2

Subtract $4 x$ from $x$ .

$\frac{2 (x^{2} - 3 x - 4 - (x - 1) (x - 2))}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.4

Apply the distributive property.

$\frac{2 (x^{2} - 3 x - 4 + (- x - - 1) (x - 2))}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.5

Multiply $- 1$ by $- 1$ .

$\frac{2 (x^{2} - 3 x - 4 + (- x + 1) (x - 2))}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.6

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

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Step 2.10.6.1

Apply the distributive property.

$\frac{2 (x^{2} - 3 x - 4 - x (x - 2) + 1 (x - 2))}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.6.2

Apply the distributive property.

$\frac{2 (x^{2} - 3 x - 4 - x \cdot x - x \cdot - 2 + 1 (x - 2))}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.6.3

Apply the distributive property.

$\frac{2 (x^{2} - 3 x - 4 - x \cdot x - x \cdot - 2 + 1 x + 1 \cdot - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7

Simplify and combine like terms.

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Step 2.10.7.1

Simplify each term.

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Step 2.10.7.1.1

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

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Step 2.10.7.1.1.1

Move $x$ .

$\frac{2 (x^{2} - 3 x - 4 - (x \cdot x) - x \cdot - 2 + 1 x + 1 \cdot - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7.1.1.2

Multiply $x$ by $x$ .

$\frac{2 (x^{2} - 3 x - 4 - x^{2} - x \cdot - 2 + 1 x + 1 \cdot - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7.1.2

Multiply $- 2$ by $- 1$ .

$\frac{2 (x^{2} - 3 x - 4 - x^{2} + 2 x + 1 x + 1 \cdot - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7.1.3

Multiply $x$ by $1$ .

$\frac{2 (x^{2} - 3 x - 4 - x^{2} + 2 x + x + 1 \cdot - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7.1.4

Multiply $- 2$ by $1$ .

$\frac{2 (x^{2} - 3 x - 4 - x^{2} + 2 x + x - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.7.2

Add $2 x$ and $x$ .

$\frac{2 (x^{2} - 3 x - 4 - x^{2} + 3 x - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.8

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

$\frac{2 (- 3 x - 4 + 0 + 3 x - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.9

Add $- 3 x$ and $0$ .

$\frac{2 (- 3 x - 4 + 3 x - 2)}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.10.10

Add $- 3 x$ and $3 x$ .

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

Step 2.10.11

Subtract $4$ from $0$ .

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

Step 2.10.12

Subtract $2$ from $- 4$ .

$\frac{2 \cdot - 6}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.11

Multiply $2$ by $- 6$ .

$\frac{- 12}{(x + 1) (x - 1) (x - 2)} = 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{12}{(x + 1) (x - 1) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$12 = 0$

Step 4

Since $12 \neq 0$ , there are no solutions.

No solution