Pre-Algebra Examples

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

Step 1

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 2

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 1, 1$

Step 2.2.2

Remove parentheses.

$x - 1, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2.3

Multiply each term in $\frac{x}{x - 1} = 2$ by $x - 1$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{x}{x - 1} = 2$ by $x - 1$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x - 1$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

$x = 2 (x - 1)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$x = 2 x + 2 \cdot - 1$

Step 2.3.3.2

Multiply $2$ by $- 1$ .

$x = 2 x - 2$

Step 2.4

Solve the equation.

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Step 2.4.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.4.1.1

Subtract $2 x$ from both sides of the equation.

$x - 2 x = - 2$

Step 2.4.1.2

Subtract $2 x$ from $x$ .

$- x = - 2$

Step 2.4.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 2.4.2.1

Divide each term in $- x = - 2$ by $- 1$ .

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Dividing two negative values results in a positive value.

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

Step 2.4.2.2.2

Divide $x$ by $1$ .

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

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 2.5

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 2.6

Find the LCD of the terms in the equation.

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Step 2.6.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 1, 1$

Step 2.6.2

Remove parentheses.

$x - 1, 1$

Step 2.6.3

The LCM of one and any expression is the expression.

$x - 1$

Step 2.7

Multiply each term in $\frac{x}{x - 1} = - 2$ by $x - 1$ to eliminate the fractions.

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Step 2.7.1

Multiply each term in $\frac{x}{x - 1} = - 2$ by $x - 1$ .

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

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

Cancel the common factor of $x - 1$ .

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Step 2.7.2.1.1

Cancel the common factor.

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

Step 2.7.2.1.2

Rewrite the expression.

$x = - 2 (x - 1)$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

Apply the distributive property.

$x = - 2 x - 2 \cdot - 1$

Step 2.7.3.2

Multiply $- 2$ by $- 1$ .

$x = - 2 x + 2$

Step 2.8

Solve the equation.

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Step 2.8.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.8.1.1

Add $2 x$ to both sides of the equation.

$x + 2 x = 2$

Step 2.8.1.2

Add $x$ and $2 x$ .

$3 x = 2$

Step 2.8.2

Divide each term in $3 x = 2$ by $3$ and simplify.

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Step 2.8.2.1

Divide each term in $3 x = 2$ by $3$ .

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

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $3$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

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

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, \frac{2}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 2, \frac{2}{3}$

Decimal Form:

$x = 2, 0.\overline{6}$