Precalculus Examples

$- 4 | x + 2 | = x - 8$

Step 1

Divide each term in $- 4 | x + 2 | = x - 8$ by $- 4$ and simplify.

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Step 1.1

Divide each term in $- 4 | x + 2 | = x - 8$ by $- 4$ .

$\frac{- 4 | x + 2 |}{- 4} = \frac{x}{- 4} + \frac{- 8}{- 4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 4 | x + 2 |}{- 4} = \frac{x}{- 4} + \frac{- 8}{- 4}$

Step 1.2.1.2

Divide $| x + 2 |$ by $1$ .

$| x + 2 | = \frac{x}{- 4} + \frac{- 8}{- 4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Move the negative in front of the fraction.

$| x + 2 | = - \frac{x}{4} + \frac{- 8}{- 4}$

Step 1.3.1.2

Divide $- 8$ by $- 4$ .

$| x + 2 | = - \frac{x}{4} + 2$

Step 2

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

$x + 2 = \pm (- \frac{x}{4} + 2)$

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.

$x + 2 = - \frac{x}{4} + 2$

Step 3.2

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

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Step 3.2.1

Add $\frac{x}{4}$ to both sides of the equation.

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

Step 3.2.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x \cdot \frac{4}{4} + \frac{x}{4} + 2 = 2$

Step 3.2.3

Combine $x$ and $\frac{4}{4}$ .

$\frac{x \cdot 4}{4} + \frac{x}{4} + 2 = 2$

Step 3.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 + x}{4} + 2 = 2$

Step 3.2.5

Add $x \cdot 4$ and $x$ .

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Step 3.2.5.1

Reorder $x$ and $4$ .

$\frac{4 \cdot x + x}{4} + 2 = 2$

Step 3.2.5.2

Add $4 \cdot x$ and $x$ .

$\frac{5 \cdot x}{4} + 2 = 2$

$\frac{5 x}{4} + 2 = 2$

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.

$\frac{5 x}{4} = 2 - 2$

Step 3.3.2

Subtract $2$ from $2$ .

$\frac{5 x}{4} = 0$

Step 3.4

Set the numerator equal to zero.

$5 x = 0$

Step 3.5

Divide each term in $5 x = 0$ by $5$ and simplify.

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Step 3.5.1

Divide each term in $5 x = 0$ by $5$ .

$\frac{5 x}{5} = \frac{0}{5}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $5$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{0}{5}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{5}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide $0$ by $5$ .

$x = 0$

Step 3.6

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

$x + 2 = - (- \frac{x}{4} + 2)$

Step 3.7

Simplify $- (- \frac{x}{4} + 2)$ .

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Step 3.7.1

Rewrite.

$x + 2 = 0 + 0 - (- \frac{x}{4} + 2)$

Step 3.7.2

Simplify by adding zeros.

$x + 2 = - (- \frac{x}{4} + 2)$

Step 3.7.3

Apply the distributive property.

$x + 2 = - - \frac{x}{4} - 1 \cdot 2$

Step 3.7.4

Multiply $- - \frac{x}{4}$ .

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Step 3.7.4.1

Multiply $- 1$ by $- 1$ .

$x + 2 = 1 \frac{x}{4} - 1 \cdot 2$

Step 3.7.4.2

Multiply $\frac{x}{4}$ by $1$ .

$x + 2 = \frac{x}{4} - 1 \cdot 2$

Step 3.7.5

Multiply $- 1$ by $2$ .

$x + 2 = \frac{x}{4} - 2$

Step 3.8

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

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Step 3.8.1

Subtract $\frac{x}{4}$ from both sides of the equation.

$x + 2 - \frac{x}{4} = - 2$

Step 3.8.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x \cdot \frac{4}{4} - \frac{x}{4} + 2 = - 2$

Step 3.8.3

Combine $x$ and $\frac{4}{4}$ .

$\frac{x \cdot 4}{4} - \frac{x}{4} + 2 = - 2$

Step 3.8.4

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - x}{4} + 2 = - 2$

Step 3.8.5

Subtract $x$ from $x \cdot 4$ .

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Step 3.8.5.1

Reorder $x$ and $4$ .

$\frac{4 \cdot x - x}{4} + 2 = - 2$

Step 3.8.5.2

Subtract $x$ from $4 \cdot x$ .

$\frac{3 \cdot x}{4} + 2 = - 2$

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

Step 3.9

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

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Step 3.9.1

Subtract $2$ from both sides of the equation.

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

Step 3.9.2

Subtract $2$ from $- 2$ .

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

Step 3.10

Multiply both sides of the equation by $\frac{4}{3}$ .

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot - 4$

Step 3.11

Simplify both sides of the equation.

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Step 3.11.1

Simplify the left side.

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Step 3.11.1.1

Simplify $\frac{4}{3} \cdot \frac{3 x}{4}$ .

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Step 3.11.1.1.1

Cancel the common factor of $4$ .

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Step 3.11.1.1.1.1

Cancel the common factor.

$\frac{4}{3} \cdot \frac{3 x}{4} = \frac{4}{3} \cdot - 4$

Step 3.11.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot - 4$

Step 3.11.1.1.2

Cancel the common factor of $3$ .

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Step 3.11.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{4}{3} \cdot - 4$

Step 3.11.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{4}{3} \cdot - 4$

Step 3.11.1.1.2.3

Rewrite the expression.

$x = \frac{4}{3} \cdot - 4$

Step 3.11.2

Simplify the right side.

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Step 3.11.2.1

Simplify $\frac{4}{3} \cdot - 4$ .

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Step 3.11.2.1.1

Multiply $\frac{4}{3} \cdot - 4$ .

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Step 3.11.2.1.1.1

Combine $\frac{4}{3}$ and $- 4$ .

$x = \frac{4 \cdot - 4}{3}$

Step 3.11.2.1.1.2

Multiply $4$ by $- 4$ .

$x = \frac{- 16}{3}$

Step 3.11.2.1.2

Move the negative in front of the fraction.

$x = - \frac{16}{3}$

Step 3.12

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

$x = 0, - \frac{16}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, - \frac{16}{3}$

Decimal Form:

$x = 0, - 5.\overline{3}$

Mixed Number Form:

$x = 0, - 5 \frac{1}{3}$