Algebra Examples

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

Step 1

Convert $1 \frac{1}{2}$ to an improper fraction.

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Step 1.1

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

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

Step 1.2

Add $1$ and $\frac{1}{2}$ .

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Step 1.2.1

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

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.2.3

Add $2$ and $1$ .

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

Step 2

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

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

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.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Combine the numerators over the common denominator.

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

Step 3.2.3

Add $3$ and $1$ .

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

Step 3.2.4

Divide $4$ by $2$ .

$3 x = 2$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $3$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

Combine the numerators over the common denominator.

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

Step 3.5.3

Add $- 3$ and $1$ .

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

Step 3.5.4

Divide $- 2$ by $2$ .

$3 x = - 1$

Step 3.6

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 3.6.1

Divide each term in $3 x = - 1$ by $3$ .

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

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Cancel the common factor of $3$ .

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Step 3.6.2.1.1

Cancel the common factor.

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

Step 3.6.2.1.2

Divide $x$ by $1$ .

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

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Move the negative in front of the fraction.

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

Step 3.7

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{6}, - 0.\overline{3}$