Precalculus Examples

$| 2 - 4 x | = 12$

Step 1

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

$2 - 4 x = \pm 12$

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.

$2 - 4 x = 12$

Step 2.2

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

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Step 2.2.1

Subtract $2$ from both sides of the equation.

$- 4 x = 12 - 2$

Step 2.2.2

Subtract $2$ from $12$ .

$- 4 x = 10$

Step 2.3

Divide each term in $- 4 x = 10$ by $- 4$ and simplify.

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Step 2.3.1

Divide each term in $- 4 x = 10$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{10}{- 4}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 4$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{10}{- 4}$

Step 2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{- 4}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of $10$ and $- 4$ .

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Step 2.3.3.1.1

Factor $2$ out of $10$ .

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

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor $2$ out of $- 4$ .

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

Step 2.3.3.1.2.2

Cancel the common factor.

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

Step 2.3.3.1.2.3

Rewrite the expression.

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 2.4

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

$2 - 4 x = - 12$

Step 2.5

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

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Step 2.5.1

Subtract $2$ from both sides of the equation.

$- 4 x = - 12 - 2$

Step 2.5.2

Subtract $2$ from $- 12$ .

$- 4 x = - 14$

Step 2.6

Divide each term in $- 4 x = - 14$ by $- 4$ and simplify.

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Step 2.6.1

Divide each term in $- 4 x = - 14$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- 14}{- 4}$

Step 2.6.2

Simplify the left side.

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Step 2.6.2.1

Cancel the common factor of $- 4$ .

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Step 2.6.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- 14}{- 4}$

Step 2.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 14}{- 4}$

Step 2.6.3

Simplify the right side.

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Step 2.6.3.1

Cancel the common factor of $- 14$ and $- 4$ .

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Step 2.6.3.1.1

Factor $- 2$ out of $- 14$ .

$x = \frac{- 2 (7)}{- 4}$

Step 2.6.3.1.2

Cancel the common factors.

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Step 2.6.3.1.2.1

Factor $- 2$ out of $- 4$ .

$x = \frac{- 2 \cdot 7}{- 2 \cdot 2}$

Step 2.6.3.1.2.2

Cancel the common factor.

$x = \frac{- 2 \cdot 7}{- 2 \cdot 2}$

Step 2.6.3.1.2.3

Rewrite the expression.

$x = \frac{7}{2}$

Step 2.7

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

$x = - \frac{5}{2}, \frac{7}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{2}, \frac{7}{2}$

Decimal Form:

$x = - 2.5, 3.5$

Mixed Number Form:

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