Precalculus Examples

$4 | x + \frac{1}{3} | = 20$

Step 1

Divide each term in $4 | x + \frac{1}{3} | = 20$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 | x + \frac{1}{3} | = 20$ by $4$ .

$\frac{4 | x + \frac{1}{3} |}{4} = \frac{20}{4}$

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.

$\frac{4 | x + \frac{1}{3} |}{4} = \frac{20}{4}$

Step 1.2.1.2

Divide $| x + \frac{1}{3} |$ by $1$ .

$| x + \frac{1}{3} | = \frac{20}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $20$ by $4$ .

$| x + \frac{1}{3} | = 5$

Step 2

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

$x + \frac{1}{3} = \pm 5$

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.

$x + \frac{1}{3} = 5$

Step 3.2

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

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Step 3.2.1

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 3.2.2

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = 5 \cdot \frac{3}{3} - \frac{1}{3}$

Step 3.2.3

Combine $5$ and $\frac{3}{3}$ .

$x = \frac{5 \cdot 3}{3} - \frac{1}{3}$

Step 3.2.4

Combine the numerators over the common denominator.

$x = \frac{5 \cdot 3 - 1}{3}$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $5$ by $3$ .

$x = \frac{15 - 1}{3}$

Step 3.2.5.2

Subtract $1$ from $15$ .

$x = \frac{14}{3}$

Step 3.3

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

$x + \frac{1}{3} = - 5$

Step 3.4

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

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Step 3.4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

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

Step 3.4.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - 5 \cdot \frac{3}{3} - \frac{1}{3}$

Step 3.4.3

Combine $- 5$ and $\frac{3}{3}$ .

$x = \frac{- 5 \cdot 3}{3} - \frac{1}{3}$

Step 3.4.4

Combine the numerators over the common denominator.

$x = \frac{- 5 \cdot 3 - 1}{3}$

Step 3.4.5

Simplify the numerator.

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Step 3.4.5.1

Multiply $- 5$ by $3$ .

$x = \frac{- 15 - 1}{3}$

Step 3.4.5.2

Subtract $1$ from $- 15$ .

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

Step 3.4.6

Move the negative in front of the fraction.

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

Step 3.5

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

$x = \frac{14}{3}, - \frac{16}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{14}{3}, - \frac{16}{3}$

Decimal Form:

$x = 4.\overline{6}, - 5.\overline{3}$

Mixed Number Form:

$x = 4 \frac{2}{3}, - 5 \frac{1}{3}$