Precalculus Examples

$\frac{1}{4} x + \frac{1}{2} = \frac{1}{8} x - \frac{1}{3}$

Step 1

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} + \frac{1}{2} = \frac{1}{8} x - \frac{1}{3}$

Step 2

Combine $\frac{1}{8}$ and $x$ .

$\frac{x}{4} + \frac{1}{2} = \frac{x}{8} - \frac{1}{3}$

Step 3

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

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Step 3.1

Subtract $\frac{x}{8}$ from both sides of the equation.

$\frac{x}{4} + \frac{1}{2} - \frac{x}{8} = - \frac{1}{3}$

Step 3.2

To write $\frac{x}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{4} \cdot \frac{2}{2} - \frac{x}{8} + \frac{1}{2} = - \frac{1}{3}$

Step 3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

Multiply $\frac{x}{4}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{4 \cdot 2} - \frac{x}{8} + \frac{1}{2} = - \frac{1}{3}$

Step 3.3.2

Multiply $4$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Subtract $x$ from $x \cdot 2$ .

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Step 3.5.1

Reorder $x$ and $2$ .

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

Step 3.5.2

Subtract $x$ from $2 \cdot x$ .

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

Step 4

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

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Step 4.1

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Step 4.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.1

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

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

Step 4.4.2

Multiply $3$ by $2$ .

$\frac{x}{8} = - \frac{2}{6} - \frac{1}{2} \cdot \frac{3}{3}$

Step 4.4.3

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

$\frac{x}{8} = - \frac{2}{6} - \frac{3}{2 \cdot 3}$

Step 4.4.4

Multiply $2$ by $3$ .

$\frac{x}{8} = - \frac{2}{6} - \frac{3}{6}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{x}{8} = \frac{- 2 - 3}{6}$

Step 4.6

Subtract $3$ from $- 2$ .

$\frac{x}{8} = \frac{- 5}{6}$

Step 4.7

Move the negative in front of the fraction.

$\frac{x}{8} = - \frac{5}{6}$

Step 5

Multiply both sides of the equation by $8$ .

$8 \frac{x}{8} = 8 (- \frac{5}{6})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $8$ .

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Step 6.1.1.1

Cancel the common factor.

$8 \frac{x}{8} = 8 (- \frac{5}{6})$

Step 6.1.1.2

Rewrite the expression.

$x = 8 (- \frac{5}{6})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Simplify $8 (- \frac{5}{6})$ .

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Step 6.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.1.1.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$x = 8 (\frac{- 5}{6})$

Step 6.2.1.1.2

Factor $2$ out of $8$ .

$x = 2 (4) \frac{- 5}{6}$

Step 6.2.1.1.3

Factor $2$ out of $6$ .

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

Step 6.2.1.1.4

Cancel the common factor.

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

Step 6.2.1.1.5

Rewrite the expression.

$x = 4 (\frac{- 5}{3})$

Step 6.2.1.2

Combine $4$ and $\frac{- 5}{3}$ .

$x = \frac{4 \cdot - 5}{3}$

Step 6.2.1.3

Simplify the expression.

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Step 6.2.1.3.1

Multiply $4$ by $- 5$ .

$x = \frac{- 20}{3}$

Step 6.2.1.3.2

Move the negative in front of the fraction.

$x = - \frac{20}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{20}{3}$

Decimal Form:

$x = - 6.\overline{6}$

Mixed Number Form:

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