Pre-Algebra Examples

$| 3 x - 2 | = 8 - x$

Step 1

Reorder $8$ and $- x$ .

$| 3 x - 2 | = - x + 8$

Step 2

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

$3 x - 2 = \pm (- x + 8)$

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.

$3 x - 2 = - x + 8$

Step 3.2

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

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Step 3.2.1

Add $x$ to both sides of the equation.

$3 x - 2 + x = 8$

Step 3.2.2

Add $3 x$ and $x$ .

$4 x - 2 = 8$

Step 3.3

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

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Step 3.3.1

Add $2$ to both sides of the equation.

$4 x = 8 + 2$

Step 3.3.2

Add $8$ and $2$ .

$4 x = 10$

Step 3.4

Divide each term in $4 x = 10$ by $4$ and simplify.

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Step 3.4.1

Divide each term in $4 x = 10$ by $4$ .

$\frac{4 x}{4} = \frac{10}{4}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $4$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{10}{4}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{4}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $10$ and $4$ .

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Step 3.4.3.1.1

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{4}$

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 5}{2 \cdot 2}$

Step 3.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 5}{2 \cdot 2}$

Step 3.4.3.1.2.3

Rewrite the expression.

$x = \frac{5}{2}$

Step 3.5

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

$3 x - 2 = - (- x + 8)$

Step 3.6

Simplify $- (- x + 8)$ .

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Step 3.6.1

Rewrite.

$3 x - 2 = 0 + 0 - (- x + 8)$

Step 3.6.2

Simplify by adding zeros.

$3 x - 2 = - (- x + 8)$

Step 3.6.3

Apply the distributive property.

$3 x - 2 = - - x - 1 \cdot 8$

Step 3.6.4

Multiply $- - x$ .

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Step 3.6.4.1

Multiply $- 1$ by $- 1$ .

$3 x - 2 = 1 x - 1 \cdot 8$

Step 3.6.4.2

Multiply $x$ by $1$ .

$3 x - 2 = x - 1 \cdot 8$

Step 3.6.5

Multiply $- 1$ by $8$ .

$3 x - 2 = x - 8$

Step 3.7

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

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Step 3.7.1

Subtract $x$ from both sides of the equation.

$3 x - 2 - x = - 8$

Step 3.7.2

Subtract $x$ from $3 x$ .

$2 x - 2 = - 8$

Step 3.8

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

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Step 3.8.1

Add $2$ to both sides of the equation.

$2 x = - 8 + 2$

Step 3.8.2

Add $- 8$ and $2$ .

$2 x = - 6$

Step 3.9

Divide each term in $2 x = - 6$ by $2$ and simplify.

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Step 3.9.1

Divide each term in $2 x = - 6$ by $2$ .

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

Step 3.9.2

Simplify the left side.

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Step 3.9.2.1

Cancel the common factor of $2$ .

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Step 3.9.2.1.1

Cancel the common factor.

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

Step 3.9.2.1.2

Divide $x$ by $1$ .

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

Step 3.9.3

Simplify the right side.

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Step 3.9.3.1

Divide $- 6$ by $2$ .

$x = - 3$

Step 3.10

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.5, - 3$

Mixed Number Form:

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