Precalculus Examples

- 4 | x + 2 | = x - 8

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

Step 1

Divide each term in

- 4 | x + 2 | = x - 8

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

- 4

$- 4$ and simplify.

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Step 1.1

Divide each term in

- 4 | x + 2 | = x - 8

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

- 4

$- 4$ .

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

$- 4$ .

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Step 1.2.1.1

Cancel the common factor.

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

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

Step 1.2.1.2

Divide

| x + 2 |

$| x + 2 |$ by

1

$1$ .

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

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

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

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

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

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

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}

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

Step 1.3.1.2

Divide

- 8

$- 8$ by

- 4

$- 4$ .

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

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

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

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

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

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

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

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

Step 2

Remove the absolute value term. This creates a

\pm

$\pm$ on the right side of the equation because

| x | = \pm x

$| x | = \pm x$ .

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

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

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

$\pm$ to find the first solution.

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

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

Step 3.2

Move all terms containing

x

$x$ to the left side of the equation.

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Step 3.2.1

Add

\frac{x}{4}

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

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

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

Step 3.2.2

To write

x

$x$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 3.2.3

Combine

x

$x$ and

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 3.2.4

Combine the numerators over the common denominator.

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

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

Step 3.2.5

Add

x \cdot 4

$x \cdot 4$ and

x

$x$ .

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Step 3.2.5.1

Reorder

x

$x$ and

4

$4$ .

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

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

Step 3.2.5.2

Add

4 \cdot x

$4 \cdot x$ and

x

$x$ .

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

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

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

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

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

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

Step 3.3

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 3.3.1

Subtract

2

$2$ from both sides of the equation.

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

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

Step 3.3.2

Subtract

2

$2$ from

2

$2$ .

\frac{5 x}{4} = 0

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

\frac{5 x}{4} = 0

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

Step 3.4

Set the numerator equal to zero.

5 x = 0

$5 x = 0$

Step 3.5

Divide each term in

5 x = 0

$5 x = 0$ by

5

$5$ and simplify.

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Step 3.5.1

Divide each term in

5 x = 0

$5 x = 0$ by

5

$5$ .

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

$5$ .

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Step 3.5.2.1.1

Cancel the common factor.

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

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

Step 3.5.2.1.2

Divide

x

$x$ by

1

$1$ .

x = \frac{0}{5}

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

x = \frac{0}{5}

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

x = \frac{0}{5}

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

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Divide

0

$0$ by

5

$5$ .

x = 0

$x = 0$

x = 0

$x = 0$

x = 0

$x = 0$

Step 3.6

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

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

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

Step 3.7

Simplify

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

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

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Step 3.7.1

Rewrite.

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

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

Step 3.7.2

Simplify by adding zeros.

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

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

Step 3.7.3

Apply the distributive property.

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

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

Step 3.7.4

Multiply

- - \frac{x}{4}

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

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Step 3.7.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

Step 3.7.4.2

Multiply

\frac{x}{4}

$\frac{x}{4}$ by

1

$1$ .

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

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

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

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

Step 3.7.5

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

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

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

Step 3.8

Move all terms containing

x

$x$ to the left side of the equation.

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Step 3.8.1

Subtract

\frac{x}{4}

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

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

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

Step 3.8.2

To write

x

$x$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 3.8.3

Combine

x

$x$ and

\frac{4}{4}

$\frac{4}{4}$ .

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

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

Step 3.8.4

Combine the numerators over the common denominator.

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

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

Step 3.8.5

Subtract

x

$x$ from

x \cdot 4

$x \cdot 4$ .

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Step 3.8.5.1

Reorder

x

$x$ and

4

$4$ .

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

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

Step 3.8.5.2

Subtract

x

$x$ from

4 \cdot x

$4 \cdot x$ .

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

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

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

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

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

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

Step 3.9

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 3.9.1

Subtract

2

$2$ from both sides of the equation.

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

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

Step 3.9.2

Subtract

2

$2$ from

- 2

$- 2$ .

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

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

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

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

Step 3.10

Multiply both sides of the equation by

\frac{4}{3}

$\frac{4}{3}$ .

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

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

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}

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

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Step 3.11.1.1.1

Cancel the common factor of

4

$4$ .

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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

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

Step 3.11.1.1.1.2

Rewrite the expression.

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

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

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

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

Step 3.11.1.1.2

Cancel the common factor of

3

$3$ .

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Step 3.11.1.1.2.1

Factor

3

$3$ out of

3 x

$3 x$ .

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

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

Step 3.11.1.1.2.2

Cancel the common factor.

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

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

Step 3.11.1.1.2.3

Rewrite the expression.

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

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

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

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

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

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

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

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

Step 3.11.2

Simplify the right side.

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Step 3.11.2.1

Simplify

\frac{4}{3} \cdot - 4

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

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Step 3.11.2.1.1

Multiply

\frac{4}{3} \cdot - 4

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

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Step 3.11.2.1.1.1

Combine

\frac{4}{3}

$\frac{4}{3}$ and

- 4

$- 4$ .

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

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

Step 3.11.2.1.1.2

Multiply

4

$4$ by

- 4

$- 4$ .

x = \frac{- 16}{3}

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

x = \frac{- 16}{3}

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

Step 3.11.2.1.2

Move the negative in front of the fraction.

x = - \frac{16}{3}

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

x = - \frac{16}{3}

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

x = - \frac{16}{3}

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

x = - \frac{16}{3}

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

Step 3.12

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

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

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

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

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

Decimal Form:

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

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

Mixed Number Form:

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

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