Pre-Algebra Examples

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

Step 1

Move all terms not containing $| 2 - \frac{2}{3} x |$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$| 2 - \frac{2}{3} x | = 5 + 3$

Step 1.2

Add $5$ and $3$ .

$| 2 - \frac{2}{3} x | = 8$

Step 2

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

$2 - \frac{2}{3} x = \pm 8$

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.

$2 - \frac{2}{3} x = 8$

Step 3.2

Simplify each term.

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Step 3.2.1

Combine $x$ and $\frac{2}{3}$ .

$2 - \frac{x \cdot 2}{3} = 8$

Step 3.2.2

Move $2$ to the left of $x$ .

$2 - \frac{2 x}{3} = 8$

Step 3.3

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

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Step 3.3.1

Subtract $2$ from both sides of the equation.

$- \frac{2 x}{3} = 8 - 2$

Step 3.3.2

Subtract $2$ from $8$ .

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

Step 3.4

Multiply both sides of the equation by $- \frac{3}{2}$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $- \frac{3}{2} (- \frac{2 x}{3})$ .

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Step 3.5.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.5.1.1.1.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

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

Step 3.5.1.1.1.3

Factor $3$ out of $- 3$ .

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

Step 3.5.1.1.1.4

Cancel the common factor.

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

Step 3.5.1.1.1.5

Rewrite the expression.

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

Step 3.5.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.1.1.2.1

Factor $2$ out of $- 2 x$ .

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

Step 3.5.1.1.2.2

Cancel the common factor.

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

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.1.1.3

Multiply.

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Step 3.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.5.1.1.3.2

Multiply $x$ by $1$ .

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $- \frac{3}{2} \cdot 6$ .

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Step 3.5.2.1.1

Cancel the common factor of $2$ .

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Step 3.5.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 3.5.2.1.1.2

Factor $2$ out of $6$ .

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

Step 3.5.2.1.1.3

Cancel the common factor.

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

Step 3.5.2.1.1.4

Rewrite the expression.

$x = - 3 \cdot 3$

Step 3.5.2.1.2

Multiply $- 3$ by $3$ .

$x = - 9$

Step 3.6

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

$2 - \frac{2}{3} x = - 8$

Step 3.7

Simplify each term.

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Step 3.7.1

Combine $x$ and $\frac{2}{3}$ .

$2 - \frac{x \cdot 2}{3} = - 8$

Step 3.7.2

Move $2$ to the left of $x$ .

$2 - \frac{2 x}{3} = - 8$

Step 3.8

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

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Step 3.8.1

Subtract $2$ from both sides of the equation.

$- \frac{2 x}{3} = - 8 - 2$

Step 3.8.2

Subtract $2$ from $- 8$ .

$- \frac{2 x}{3} = - 10$

Step 3.9

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 x}{3}) = - \frac{3}{2} \cdot - 10$

Step 3.10

Simplify both sides of the equation.

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Step 3.10.1

Simplify the left side.

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Step 3.10.1.1

Simplify $- \frac{3}{2} (- \frac{2 x}{3})$ .

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Step 3.10.1.1.1

Cancel the common factor of $3$ .

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Step 3.10.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 x}{3}) = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.1.2

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 x}{3} = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 x}{3} = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 x}{3} = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 x) = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.2

Cancel the common factor of $2$ .

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Step 3.10.1.1.2.1

Factor $2$ out of $- 2 x$ .

$\frac{- 1}{2} (2 (- x)) = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- x)) = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.3

Multiply.

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Step 3.10.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{2} \cdot - 10$

Step 3.10.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{2} \cdot - 10$

Step 3.10.2

Simplify the right side.

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Step 3.10.2.1

Simplify $- \frac{3}{2} \cdot - 10$ .

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Step 3.10.2.1.1

Cancel the common factor of $2$ .

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Step 3.10.2.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$x = \frac{- 3}{2} \cdot - 10$

Step 3.10.2.1.1.2

Factor $2$ out of $- 10$ .

$x = \frac{- 3}{2} \cdot (2 (- 5))$

Step 3.10.2.1.1.3

Cancel the common factor.

$x = \frac{- 3}{2} \cdot (2 \cdot - 5)$

Step 3.10.2.1.1.4

Rewrite the expression.

$x = - 3 \cdot - 5$

Step 3.10.2.1.2

Multiply $- 3$ by $- 5$ .

$x = 15$

Step 3.11

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

$x = - 9, 15$