Algebra Examples

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

Step 1

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

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

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.

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

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 $\frac{1}{2}$ to both sides of the equation.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $2$ .

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

Step 2.2.5.2

Add $4$ and $1$ .

$x = \frac{5}{2}$

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

Add $\frac{1}{2}$ to both sides of the equation.

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

Combine the numerators over the common denominator.

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

Step 2.4.5

Simplify the numerator.

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Step 2.4.5.1

Multiply $- 2$ by $2$ .

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

Step 2.4.5.2

Add $- 4$ and $1$ .

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

Step 2.4.6

Move the negative in front of the fraction.

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

Step 2.5

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

$x = \frac{5}{2}, - \frac{3}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}, - \frac{3}{2}$

Decimal Form:

$x = 2.5, - 1.5$

Mixed Number Form:

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