Algebra Examples

$| \frac{x}{2} - \frac{1}{3} | = 1$

Step 1

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

$\frac{x}{2} - \frac{1}{3} = \pm 1$

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.

$\frac{x}{2} - \frac{1}{3} = 1$

Step 2.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.2.1

Add $\frac{1}{3}$ to both sides of the equation.

$\frac{x}{2} = 1 + \frac{1}{3}$

Step 2.2.2

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

$\frac{x}{2} = \frac{3}{3} + \frac{1}{3}$

Step 2.2.3

Combine the numerators over the common denominator.

$\frac{x}{2} = \frac{3 + 1}{3}$

Step 2.2.4

Add $3$ and $1$ .

$\frac{x}{2} = \frac{4}{3}$

Step 2.3

Multiply both sides of the equation by $2$ .

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

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1

Cancel the common factor.

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

Step 2.4.1.1.2

Rewrite the expression.

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $2 (\frac{4}{3})$ .

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Step 2.4.2.1.1

Combine $2$ and $\frac{4}{3}$ .

$x = \frac{2 \cdot 4}{3}$

Step 2.4.2.1.2

Multiply $2$ by $4$ .

$x = \frac{8}{3}$

Step 2.5

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

$\frac{x}{2} - \frac{1}{3} = - 1$

Step 2.6

Move all terms not containing $x$ to the right side of the equation.

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Step 2.6.1

Add $\frac{1}{3}$ to both sides of the equation.

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

Step 2.6.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.6.3

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 2.6.4

Combine the numerators over the common denominator.

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

Step 2.6.5

Simplify the numerator.

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Step 2.6.5.1

Multiply $- 1$ by $3$ .

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

Step 2.6.5.2

Add $- 3$ and $1$ .

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

Step 2.6.6

Move the negative in front of the fraction.

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

Step 2.7

Multiply both sides of the equation by $2$ .

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

Step 2.8

Simplify both sides of the equation.

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Step 2.8.1

Simplify the left side.

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Step 2.8.1.1

Cancel the common factor of $2$ .

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Step 2.8.1.1.1

Cancel the common factor.

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

Step 2.8.1.1.2

Rewrite the expression.

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

Step 2.8.2

Simplify the right side.

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Step 2.8.2.1

Simplify $2 (- \frac{2}{3})$ .

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Step 2.8.2.1.1

Multiply $2 (- \frac{2}{3})$ .

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Step 2.8.2.1.1.1

Multiply $- 1$ by $2$ .

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

Step 2.8.2.1.1.2

Combine $- 2$ and $\frac{2}{3}$ .

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

Step 2.8.2.1.1.3

Multiply $- 2$ by $2$ .

$x = \frac{- 4}{3}$

Step 2.8.2.1.2

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 2.9

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

$x = \frac{8}{3}, - \frac{4}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}, - \frac{4}{3}$

Decimal Form:

$x = 2.\overline{6}, - 1.\overline{3}$

Mixed Number Form:

$x = 2 \frac{2}{3}, - 1 \frac{1}{3}$