Algebra Examples

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

Step 1

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

$\frac{x}{3} - 1 = \pm (\frac{x}{6} + 2)$

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.

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

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

Subtract $\frac{x}{6}$ from both sides of the equation.

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

Multiply $3$ by $2$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Subtract $x$ from $x \cdot 2$ .

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Step 2.2.5.1

Reorder $x$ and $2$ .

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

Step 2.2.5.2

Subtract $x$ from $2 \cdot x$ .

$\frac{x}{6} - 1 = 2$

Step 2.3

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

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Step 2.3.1

Add $1$ to both sides of the equation.

$\frac{x}{6} = 2 + 1$

Step 2.3.2

Add $2$ and $1$ .

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

Step 2.4

Multiply both sides of the equation by $6$ .

$6 \frac{x}{6} = 6 \cdot 3$

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $6$ .

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Step 2.5.1.1.1

Cancel the common factor.

$6 \frac{x}{6} = 6 \cdot 3$

Step 2.5.1.1.2

Rewrite the expression.

$x = 6 \cdot 3$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Multiply $6$ by $3$ .

$x = 18$

Step 2.6

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

$\frac{x}{3} - 1 = - (\frac{x}{6} + 2)$

Step 2.7

Simplify $- (\frac{x}{6} + 2)$ .

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Step 2.7.1

Rewrite.

$\frac{x}{3} - 1 = 0 + 0 - (\frac{x}{6} + 2)$

Step 2.7.2

Simplify by adding zeros.

$\frac{x}{3} - 1 = - (\frac{x}{6} + 2)$

Step 2.7.3

Apply the distributive property.

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

Step 2.7.4

Multiply $- 1$ by $2$ .

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

Step 2.8

Move all terms containing $x$ to the left side of the equation.

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Step 2.8.1

Add $\frac{x}{6}$ to both sides of the equation.

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

Step 2.8.2

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

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

Step 2.8.3

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

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Step 2.8.3.1

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

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

Step 2.8.3.2

Multiply $3$ by $2$ .

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

Step 2.8.4

Combine the numerators over the common denominator.

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

Step 2.8.5

Add $x \cdot 2$ and $x$ .

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Step 2.8.5.1

Reorder $x$ and $2$ .

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

Step 2.8.5.2

Add $2 \cdot x$ and $x$ .

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

Step 2.8.6

Cancel the common factor of $3$ and $6$ .

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Step 2.8.6.1

Factor $3$ out of $3 \cdot x$ .

$\frac{3 (x)}{6} - 1 = - 2$

Step 2.8.6.2

Cancel the common factors.

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Step 2.8.6.2.1

Factor $3$ out of $6$ .

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

Step 2.8.6.2.2

Cancel the common factor.

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

Step 2.8.6.2.3

Rewrite the expression.

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

Step 2.9

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

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Step 2.9.1

Add $1$ to both sides of the equation.

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

Step 2.9.2

Add $- 2$ and $1$ .

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

Step 2.10

Multiply both sides of the equation by $2$ .

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

Step 2.11

Simplify both sides of the equation.

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Step 2.11.1

Simplify the left side.

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Step 2.11.1.1

Cancel the common factor of $2$ .

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Step 2.11.1.1.1

Cancel the common factor.

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

Step 2.11.1.1.2

Rewrite the expression.

$x = 2 \cdot - 1$

Step 2.11.2

Simplify the right side.

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Step 2.11.2.1

Multiply $2$ by $- 1$ .

$x = - 2$

Step 2.12

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

$x = 18, - 2$