Algebra Examples

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

Step 1

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

$\frac{1}{2} x + 3 = \pm (x + 3)$

Step 2

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

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Step 2.1

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

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

Step 2.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.3

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

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Step 2.3.1

Subtract $x$ from both sides of the equation.

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

Step 2.3.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.3.3

Combine $- x$ and $\frac{2}{2}$ .

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

Step 2.3.4

Combine the numerators over the common denominator.

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

Step 2.3.5

Simplify each term.

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

Factor $x$ out of $x - x \cdot 2$ .

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Step 2.3.5.1.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} - x \cdot 2}{2} + 3 = 3$

Step 2.3.5.1.1.2

Factor $x$ out of $x^{1}$ .

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

Step 2.3.5.1.1.3

Factor $x$ out of $- x \cdot 2$ .

$\frac{x \cdot 1 + x (- 1 \cdot 2)}{2} + 3 = 3$

Step 2.3.5.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 1 \cdot 2)$ .

$\frac{x (1 - 1 \cdot 2)}{2} + 3 = 3$

Step 2.3.5.1.2

Multiply $- 1$ by $2$ .

$\frac{x (1 - 2)}{2} + 3 = 3$

Step 2.3.5.1.3

Subtract $2$ from $1$ .

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

Step 2.3.5.2

Move $- 1$ to the left of $x$ .

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

Step 2.3.5.3

Move the negative in front of the fraction.

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

Step 2.4

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

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Step 2.4.1

Subtract $3$ from both sides of the equation.

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

Step 2.4.2

Subtract $3$ from $3$ .

$- \frac{x}{2} = 0$

Step 2.5

Set the numerator equal to zero.

$x = 0$

Step 2.6

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

$\frac{1}{2} x + 3 = - (x + 3)$

Step 2.7

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + 3 = - (x + 3)$

Step 2.8

Simplify $- (x + 3)$ .

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Step 2.8.1

Apply the distributive property.

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

Step 2.8.2

Multiply $- 1$ by $3$ .

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

Step 2.9

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

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Step 2.9.1

Add $x$ to both sides of the equation.

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

Step 2.9.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.9.3

Combine $x$ and $\frac{2}{2}$ .

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

Step 2.9.4

Combine the numerators over the common denominator.

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

Step 2.9.5

Add $x$ and $x \cdot 2$ .

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Step 2.9.5.1

Reorder $x$ and $2$ .

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

Step 2.9.5.2

Add $x$ and $2 \cdot x$ .

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

Step 2.10

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

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Step 2.10.1

Subtract $3$ from both sides of the equation.

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

Step 2.10.2

Subtract $3$ from $- 3$ .

$\frac{3 x}{2} = - 6$

Step 2.11

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot - 6$

Step 2.12

Simplify both sides of the equation.

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Step 2.12.1

Simplify the left side.

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Step 2.12.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

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Step 2.12.1.1.1

Cancel the common factor of $2$ .

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Step 2.12.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot - 6$

Step 2.12.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot - 6$

Step 2.12.1.1.2

Cancel the common factor of $3$ .

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Step 2.12.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} \cdot - 6$

Step 2.12.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot - 6$

Step 2.12.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} \cdot - 6$

Step 2.12.2

Simplify the right side.

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Step 2.12.2.1

Simplify $\frac{2}{3} \cdot - 6$ .

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Step 2.12.2.1.1

Cancel the common factor of $3$ .

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Step 2.12.2.1.1.1

Factor $3$ out of $- 6$ .

$x = \frac{2}{3} \cdot (3 (- 2))$

Step 2.12.2.1.1.2

Cancel the common factor.

$x = \frac{2}{3} \cdot (3 \cdot - 2)$

Step 2.12.2.1.1.3

Rewrite the expression.

$x = 2 \cdot - 2$

Step 2.12.2.1.2

Multiply $2$ by $- 2$ .

$x = - 4$

Step 2.13

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

$x = 0, - 4$

Step 3

Exclude the solutions that do not make $| \frac{1}{2} x + 3 | = x + 3$ true.

$x = 0$