Finite Math Examples

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

Step 1

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

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

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.

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

Step 2.2

Multiply both sides by $4$ .

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

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Simplify $\frac{1 - 2 x}{4} \cdot 4$ .

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Step 2.3.1.1.1

Cancel the common factor of $4$ .

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Step 2.3.1.1.1.1

Cancel the common factor.

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

Step 2.3.1.1.1.2

Rewrite the expression.

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

Step 2.3.1.1.2

Reorder $1$ and $- 2 x$ .

$- 2 x + 1 = \frac{2}{3} \cdot 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $\frac{2}{3} \cdot 4$ .

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Step 2.3.2.1.1

Combine $\frac{2}{3}$ and $4$ .

$- 2 x + 1 = \frac{2 \cdot 4}{3}$

Step 2.3.2.1.2

Multiply $2$ by $4$ .

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

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Subtract $1$ from both sides of the equation.

$- 2 x = \frac{8}{3} - 1$

Step 2.4.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 2 x = \frac{8}{3} - 1 \cdot \frac{3}{3}$

Step 2.4.1.3

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

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

Step 2.4.1.4

Combine the numerators over the common denominator.

$- 2 x = \frac{8 - 1 \cdot 3}{3}$

Step 2.4.1.5

Simplify the numerator.

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Step 2.4.1.5.1

Multiply $- 1$ by $3$ .

$- 2 x = \frac{8 - 3}{3}$

Step 2.4.1.5.2

Subtract $3$ from $8$ .

$- 2 x = \frac{5}{3}$

Step 2.4.2

Divide each term in $- 2 x = \frac{5}{3}$ by $- 2$ and simplify.

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Step 2.4.2.1

Divide each term in $- 2 x = \frac{5}{3}$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{\frac{5}{3}}{- 2}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{\frac{5}{3}}{- 2}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5}{3}}{- 2}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.2.3.2

Move the negative in front of the fraction.

$x = \frac{5}{3} (- \frac{1}{2})$

Step 2.4.2.3.3

Multiply $\frac{5}{3} (- \frac{1}{2})$ .

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Step 2.4.2.3.3.1

Multiply $\frac{5}{3}$ by $\frac{1}{2}$ .

$x = - \frac{5}{3 \cdot 2}$

Step 2.4.2.3.3.2

Multiply $3$ by $2$ .

$x = - \frac{5}{6}$

Step 2.5

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

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

Step 2.6

Multiply both sides by $4$ .

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

Step 2.7

Simplify.

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Step 2.7.1

Simplify the left side.

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Step 2.7.1.1

Simplify $\frac{1 - 2 x}{4} \cdot 4$ .

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Step 2.7.1.1.1

Cancel the common factor of $4$ .

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Step 2.7.1.1.1.1

Cancel the common factor.

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

Step 2.7.1.1.1.2

Rewrite the expression.

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

Step 2.7.1.1.2

Reorder $1$ and $- 2 x$ .

$- 2 x + 1 = - \frac{2}{3} \cdot 4$

Step 2.7.2

Simplify the right side.

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Step 2.7.2.1

Simplify $- \frac{2}{3} \cdot 4$ .

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Step 2.7.2.1.1

Multiply $- \frac{2}{3} \cdot 4$ .

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Step 2.7.2.1.1.1

Multiply $4$ by $- 1$ .

$- 2 x + 1 = - 4 (\frac{2}{3})$

Step 2.7.2.1.1.2

Combine $- 4$ and $\frac{2}{3}$ .

$- 2 x + 1 = \frac{- 4 \cdot 2}{3}$

Step 2.7.2.1.1.3

Multiply $- 4$ by $2$ .

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

Step 2.7.2.1.2

Move the negative in front of the fraction.

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

Step 2.8

Solve for $x$ .

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Step 2.8.1

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

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Step 2.8.1.1

Subtract $1$ from both sides of the equation.

$- 2 x = - \frac{8}{3} - 1$

Step 2.8.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 2 x = - \frac{8}{3} - 1 \cdot \frac{3}{3}$

Step 2.8.1.3

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

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

Step 2.8.1.4

Combine the numerators over the common denominator.

$- 2 x = \frac{- 8 - 1 \cdot 3}{3}$

Step 2.8.1.5

Simplify the numerator.

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Step 2.8.1.5.1

Multiply $- 1$ by $3$ .

$- 2 x = \frac{- 8 - 3}{3}$

Step 2.8.1.5.2

Subtract $3$ from $- 8$ .

$- 2 x = \frac{- 11}{3}$

Step 2.8.1.6

Move the negative in front of the fraction.

$- 2 x = - \frac{11}{3}$

Step 2.8.2

Divide each term in $- 2 x = - \frac{11}{3}$ by $- 2$ and simplify.

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Step 2.8.2.1

Divide each term in $- 2 x = - \frac{11}{3}$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- \frac{11}{3}}{- 2}$

Step 2.8.2.2

Simplify the left side.

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Step 2.8.2.2.1

Cancel the common factor of $- 2$ .

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Step 2.8.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- \frac{11}{3}}{- 2}$

Step 2.8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{11}{3}}{- 2}$

Step 2.8.2.3

Simplify the right side.

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Step 2.8.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{11}{3} \cdot \frac{1}{- 2}$

Step 2.8.2.3.2

Move the negative in front of the fraction.

$x = - \frac{11}{3} (- \frac{1}{2})$

Step 2.8.2.3.3

Multiply $- \frac{11}{3} (- \frac{1}{2})$ .

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Step 2.8.2.3.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{11}{3}) \frac{1}{2}$

Step 2.8.2.3.3.2

Multiply $\frac{11}{3}$ by $1$ .

$x = \frac{11}{3} \cdot \frac{1}{2}$

Step 2.8.2.3.3.3

Multiply $\frac{11}{3}$ by $\frac{1}{2}$ .

$x = \frac{11}{3 \cdot 2}$

Step 2.8.2.3.3.4

Multiply $3$ by $2$ .

$x = \frac{11}{6}$

Step 2.9

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

$x = - \frac{5}{6}, \frac{11}{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5}{6}, \frac{11}{6}$

Decimal Form:

$x = - 0.8\overline{3}, 1.8\overline{3}$

Mixed Number Form:

$x = - \frac{5}{6}, 1 \frac{5}{6}$