Solve for x |x-1/2|=3 2/3

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

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

Add $9$ and $2$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

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

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $11$ by $2$ .

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

Add $22$ and $3$ .

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply $3$ by $2$ .

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

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

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

Multiply $2$ by $3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 11$ by $2$ .

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

Add $- 22$ and $3$ .

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

Move the negative in front of the fraction.

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

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

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Mixed Number Form:

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

Solve for x |x-1/2|=3 2/3

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