Algebra Examples

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

Step 1

Convert $3 \frac{2}{3}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

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

Step 1.2

Add $3$ and $\frac{2}{3}$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $3$ by $3$ .

$| x + \frac{1}{2} | = \frac{9 + 2}{3}$

Step 1.2.4.2

Add $9$ and $2$ .

$| x + \frac{1}{2} | = \frac{11}{3}$

Step 2

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

$x + \frac{1}{2} = \pm \frac{11}{3}$

Step 3

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

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Step 3.1

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

$x + \frac{1}{2} = \frac{11}{3}$

Step 3.2

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

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Step 3.2.1

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

$x = \frac{11}{3} - \frac{1}{2}$

Step 3.2.2

To write $\frac{11}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{11}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 3.2.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{11}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.2.4.1

Multiply $\frac{11}{3}$ by $\frac{2}{2}$ .

$x = \frac{11 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.2.4.2

Multiply $3$ by $2$ .

$x = \frac{11 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.2.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$x = \frac{11 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 3.2.4.4

Multiply $2$ by $3$ .

$x = \frac{11 \cdot 2}{6} - \frac{3}{6}$

Step 3.2.5

Combine the numerators over the common denominator.

$x = \frac{11 \cdot 2 - 3}{6}$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $11$ by $2$ .

$x = \frac{22 - 3}{6}$

Step 3.2.6.2

Subtract $3$ from $22$ .

$x = \frac{19}{6}$

Step 3.3

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

$x + \frac{1}{2} = - \frac{11}{3}$

Step 3.4

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

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Step 3.4.1

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

$x = - \frac{11}{3} - \frac{1}{2}$

Step 3.4.2

To write $- \frac{11}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{11}{3} \cdot \frac{2}{2} - \frac{1}{2}$

Step 3.4.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{11}{3} \cdot \frac{2}{2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.4.4.1

Multiply $\frac{11}{3}$ by $\frac{2}{2}$ .

$x = - \frac{11 \cdot 2}{3 \cdot 2} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.4.4.2

Multiply $3$ by $2$ .

$x = - \frac{11 \cdot 2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 3.4.4.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$x = - \frac{11 \cdot 2}{6} - \frac{3}{2 \cdot 3}$

Step 3.4.4.4

Multiply $2$ by $3$ .

$x = - \frac{11 \cdot 2}{6} - \frac{3}{6}$

Step 3.4.5

Combine the numerators over the common denominator.

$x = \frac{- 11 \cdot 2 - 3}{6}$

Step 3.4.6

Simplify the numerator.

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Step 3.4.6.1

Multiply $- 11$ by $2$ .

$x = \frac{- 22 - 3}{6}$

Step 3.4.6.2

Subtract $3$ from $- 22$ .

$x = \frac{- 25}{6}$

Step 3.4.7

Move the negative in front of the fraction.

$x = - \frac{25}{6}$

Step 3.5

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

$x = \frac{19}{6}, - \frac{25}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19}{6}, - \frac{25}{6}$

Decimal Form:

$x = 3.1\overline{6}, - 4.1\overline{6}$

Mixed Number Form:

$x = 3 \frac{1}{6}, - 4 \frac{1}{6}$