Algebra Examples

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

Step 1

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

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Step 1.1

Subtract $\frac{1}{x - 4}$ from both sides of the equation.

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

Step 1.2

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

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

Step 2

Simplify $\frac{2}{(x - 4) (x - 2)} - \frac{1}{x - 4} - \frac{2}{x - 2}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Multiply $- 1$ by $- 2$ .

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

Step 2.3.3

Apply the distributive property.

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

Step 2.3.4

Multiply $- 2$ by $- 4$ .

$\frac{2 - x + 2 - 2 x + 8}{(x - 2) (x - 4)} = 0$

Step 2.4

Add $2$ and $2$ .

$\frac{- x + 4 - 2 x + 8}{(x - 2) (x - 4)} = 0$

Step 2.5

Subtract $2 x$ from $- x$ .

$\frac{- 3 x + 4 + 8}{(x - 2) (x - 4)} = 0$

Step 2.6

Add $4$ and $8$ .

$\frac{- 3 x + 12}{(x - 2) (x - 4)} = 0$

Step 2.7

Factor $3$ out of $- 3 x + 12$ .

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Step 2.7.1

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

$\frac{3 (- x) + 12}{(x - 2) (x - 4)} = 0$

Step 2.7.2

Factor $3$ out of $12$ .

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

Step 2.7.3

Factor $3$ out of $3 (- x) + 3 (4)$ .

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

Step 2.8

Cancel the common factor of $- x + 4$ and $x - 4$ .

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Step 2.8.1

Factor $- 1$ out of $- x$ .

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

Step 2.8.2

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 2.8.3

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

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

Step 2.8.4

Rewrite $- (x - 4)$ as $- 1 (x - 4)$ .

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

Step 2.8.5

Cancel the common factor.

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

Step 2.8.6

Rewrite the expression.

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

Step 2.9

Multiply $3$ by $- 1$ .

$\frac{- 3}{x - 2} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{3}{x - 2} = 0$

Step 3

Set the numerator equal to zero.

$3 = 0$

Step 4

Since $3 \neq 0$ , there are no solutions.

No solution