Algebra Examples

$2 | \frac{1}{3} x + 1 | - 3 = 5$

Step 1

Move all terms not containing $| \frac{1}{3} x + 1 |$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$2 | \frac{1}{3} x + 1 | = 5 + 3$

Step 1.2

Add $5$ and $3$ .

$2 | \frac{1}{3} x + 1 | = 8$

Step 2

Divide each term in $2 | \frac{1}{3} x + 1 | = 8$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 | \frac{1}{3} x + 1 | = 8$ by $2$ .

$\frac{2 | \frac{1}{3} x + 1 |}{2} = \frac{8}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 | \frac{1}{3} x + 1 |}{2} = \frac{8}{2}$

Step 2.2.1.2

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

$| \frac{1}{3} x + 1 | = \frac{8}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $8$ by $2$ .

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

Step 3

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

$\frac{1}{3} x + 1 = \pm 4$

Step 4

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

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.2

Combine $\frac{1}{3}$ and $x$ .

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

Step 4.3

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

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Step 4.3.1

Subtract $1$ from both sides of the equation.

$\frac{x}{3} = 4 - 1$

Step 4.3.2

Subtract $1$ from $4$ .

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

Step 4.4

Multiply both sides of the equation by $3$ .

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

Step 4.5

Simplify both sides of the equation.

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Step 4.5.1

Simplify the left side.

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Step 4.5.1.1

Cancel the common factor of $3$ .

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Step 4.5.1.1.1

Cancel the common factor.

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

Step 4.5.1.1.2

Rewrite the expression.

$x = 3 \cdot 3$

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Multiply $3$ by $3$ .

$x = 9$

Step 4.6

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

$\frac{1}{3} x + 1 = - 4$

Step 4.7

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + 1 = - 4$

Step 4.8

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

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Step 4.8.1

Subtract $1$ from both sides of the equation.

$\frac{x}{3} = - 4 - 1$

Step 4.8.2

Subtract $1$ from $- 4$ .

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

Step 4.9

Multiply both sides of the equation by $3$ .

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

Step 4.10

Simplify both sides of the equation.

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Step 4.10.1

Simplify the left side.

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Step 4.10.1.1

Cancel the common factor of $3$ .

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Step 4.10.1.1.1

Cancel the common factor.

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

Step 4.10.1.1.2

Rewrite the expression.

$x = 3 \cdot - 5$

Step 4.10.2

Simplify the right side.

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Step 4.10.2.1

Multiply $3$ by $- 5$ .

$x = - 15$

Step 4.11

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

$x = 9, - 15$