Algebra Examples

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

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

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

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

Step 2.5.1.3

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

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Multiply $3$ by $- 2$ .

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

Step 2.5.4

Apply the distributive property.

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

Step 2.5.5

Multiply $x$ by $x$ .

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

Step 2.5.6

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

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

Step 2.5.7

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

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

Step 2.5.8

Subtract $x$ from $3 x$ .

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

Step 2.5.9

Reorder terms.

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Factor $2$ out of $2 (x^{2} + 2 x - 6) - 2 (x - 1) (x - 2)$ .

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Step 2.9.1.1

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

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

Step 2.9.1.2

Factor $2$ out of $2 (x^{2} + 2 x - 6) + 2 (- (x - 1) (x - 2))$ .

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

Step 2.9.2

Apply the distributive property.

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

Step 2.9.3

Multiply $- 1$ by $- 1$ .

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

Step 2.9.4

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

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Step 2.9.4.1

Apply the distributive property.

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

Step 2.9.4.2

Apply the distributive property.

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

Step 2.9.4.3

Apply the distributive property.

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

Step 2.9.5

Simplify and combine like terms.

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Step 2.9.5.1

Simplify each term.

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Step 2.9.5.1.1

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

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Step 2.9.5.1.1.1

Move $x$ .

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

Step 2.9.5.1.1.2

Multiply $x$ by $x$ .

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

Step 2.9.5.1.2

Multiply $- 2$ by $- 1$ .

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

Step 2.9.5.1.3

Multiply $x$ by $1$ .

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

Step 2.9.5.1.4

Multiply $- 2$ by $1$ .

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

Step 2.9.5.2

Add $2 x$ and $x$ .

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

Step 2.9.6

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

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

Step 2.9.7

Add $2 x$ and $0$ .

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

Step 2.9.8

Add $2 x$ and $3 x$ .

$\frac{2 (5 x - 6 - 2)}{(x - 1) (x - 2)} = 0$

Step 2.9.9

Subtract $2$ from $- 6$ .

$\frac{2 (5 x - 8)}{(x - 1) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$2 (5 x - 8) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $2 (5 x - 8) = 0$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 (5 x - 8) = 0$ by $2$ .

$\frac{2 (5 x - 8)}{2} = \frac{0}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 (5 x - 8)}{2} = \frac{0}{2}$

Step 4.1.2.1.2

Divide $5 x - 8$ by $1$ .

$5 x - 8 = \frac{0}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $2$ .

$5 x - 8 = 0$

Step 4.2

Add $8$ to both sides of the equation.

$5 x = 8$

Step 4.3

Divide each term in $5 x = 8$ by $5$ and simplify.

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Step 4.3.1

Divide each term in $5 x = 8$ by $5$ .

$\frac{5 x}{5} = \frac{8}{5}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $5$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{8}{5}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{5}$