Algebra Examples

$x + \frac{8}{x - 2} = 1 + \frac{4 x}{x - 2}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

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

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Step 1.1.1

Subtract $1$ from both sides of the equation.

$x + \frac{8}{x - 2} - 1 = \frac{4 x}{x - 2}$

Step 1.1.2

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

$x + \frac{8}{x - 2} - 1 - \frac{4 x}{x - 2} = 0$

Step 1.2

Simplify $x + \frac{8}{x - 2} - 1 - \frac{4 x}{x - 2}$ .

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Step 1.2.1

Find the common denominator.

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Step 1.2.1.1

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

$\frac{x}{1} + \frac{8}{x - 2} - 1 - \frac{4 x}{x - 2} = 0$

Step 1.2.1.2

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

$\frac{x}{1} \cdot \frac{x - 2}{x - 2} + \frac{8}{x - 2} - 1 - \frac{4 x}{x - 2} = 0$

Step 1.2.1.3

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

$\frac{x (x - 2)}{x - 2} + \frac{8}{x - 2} - 1 - \frac{4 x}{x - 2} = 0$

Step 1.2.1.4

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

$\frac{x (x - 2)}{x - 2} + \frac{8}{x - 2} + \frac{- 1}{1} - \frac{4 x}{x - 2} = 0$

Step 1.2.1.5

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

$\frac{x (x - 2)}{x - 2} + \frac{8}{x - 2} + \frac{- 1}{1} \cdot \frac{x - 2}{x - 2} - \frac{4 x}{x - 2} = 0$

Step 1.2.1.6

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Simplify each term.

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Step 1.2.3.1

Apply the distributive property.

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

Step 1.2.3.2

Multiply $x$ by $x$ .

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

Step 1.2.3.3

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

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

Step 1.2.3.4

Apply the distributive property.

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

Step 1.2.3.5

Multiply $- 1$ by $- 2$ .

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

Step 1.2.4

Subtract $x$ from $- 2 x$ .

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

Step 1.2.5

Subtract $4 x$ from $- 3 x$ .

$\frac{x^{2} - 7 x + 8 + 2}{x - 2} = 0$

Step 1.2.6

Add $8$ and $2$ .

$\frac{x^{2} - 7 x + 10}{x - 2} = 0$

Step 1.2.7

Factor $x^{2} - 7 x + 10$ using the AC method.

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Step 1.2.7.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 1.2.7.2

Write the factored form using these integers.

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

Step 1.2.8

Cancel the common factor of $x - 2$ .

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Step 1.2.8.1

Cancel the common factor.

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

Step 1.2.8.2

Divide $x - 5$ by $1$ .

$x - 5 = 0$

Step 2

Add $5$ to both sides of the equation.

$x = 5$