Basic Math Examples

$| \frac{1}{2} a + \frac{1}{12} | = | \frac{1}{8} a + \frac{1}{3} |$

Step 1

Rewrite the equation as $| \frac{1}{8} a + \frac{1}{3} | = | \frac{1}{2} a + \frac{1}{12} |$ .

$| \frac{1}{8} a + \frac{1}{3} | = | \frac{1}{2} a + \frac{1}{12} |$

Step 2

Rewrite the absolute value equation as four equations without absolute value bars.

$\frac{1}{8} a + \frac{1}{3} = \frac{1}{2} a + \frac{1}{12}$

$\frac{1}{8} a + \frac{1}{3} = - (\frac{1}{2} a + \frac{1}{12})$

$- (\frac{1}{8} a + \frac{1}{3}) = \frac{1}{2} a + \frac{1}{12}$

$- (\frac{1}{8} a + \frac{1}{3}) = - (\frac{1}{2} a + \frac{1}{12})$

Step 3

After simplifying, there are only two unique equations to be solved.

$\frac{1}{8} a + \frac{1}{3} = \frac{1}{2} a + \frac{1}{12}$

$\frac{1}{8} a + \frac{1}{3} = - (\frac{1}{2} a + \frac{1}{12})$

Step 4

Solve $\frac{1}{8} a + \frac{1}{3} = \frac{1}{2} a + \frac{1}{12}$ for $a$ .

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Step 4.1

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

$\frac{a}{8} + \frac{1}{3} = \frac{1}{2} a + \frac{1}{12}$

Step 4.2

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

$\frac{a}{8} + \frac{1}{3} = \frac{a}{2} + \frac{1}{12}$

Step 4.3

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

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Step 4.3.1

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

$\frac{a}{8} + \frac{1}{3} - \frac{a}{2} = \frac{1}{12}$

Step 4.3.2

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

$\frac{a}{8} - \frac{a}{2} \cdot \frac{4}{4} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.3

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

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Step 4.3.3.1

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

$\frac{a}{8} - \frac{a \cdot 4}{2 \cdot 4} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.3.2

Multiply $2$ by $4$ .

$\frac{a}{8} - \frac{a \cdot 4}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{a - a \cdot 4}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5

Simplify each term.

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Step 4.3.5.1

Simplify the numerator.

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Step 4.3.5.1.1

Factor $a$ out of $a - a \cdot 4$ .

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Step 4.3.5.1.1.1

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

$\frac{a^{1} - a \cdot 4}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.1.1.2

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

$\frac{a \cdot 1 - a \cdot 4}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.1.1.3

Factor $a$ out of $- a \cdot 4$ .

$\frac{a \cdot 1 + a (- 1 \cdot 4)}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.1.1.4

Factor $a$ out of $a \cdot 1 + a (- 1 \cdot 4)$ .

$\frac{a (1 - 1 \cdot 4)}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.1.2

Multiply $- 1$ by $4$ .

$\frac{a (1 - 4)}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.1.3

Subtract $4$ from $1$ .

$\frac{a \cdot - 3}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.2

Move $- 3$ to the left of $a$ .

$\frac{- 3 \cdot a}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.3.5.3

Move the negative in front of the fraction.

$- \frac{3 a}{8} + \frac{1}{3} = \frac{1}{12}$

Step 4.4

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

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Step 4.4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{3 a}{8} = \frac{1}{12} - \frac{1}{3}$

Step 4.4.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{3 a}{8} = \frac{1}{12} - \frac{1}{3} \cdot \frac{4}{4}$

Step 4.4.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.3.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$- \frac{3 a}{8} = \frac{1}{12} - \frac{4}{3 \cdot 4}$

Step 4.4.3.2

Multiply $3$ by $4$ .

$- \frac{3 a}{8} = \frac{1}{12} - \frac{4}{12}$

Step 4.4.4

Combine the numerators over the common denominator.

$- \frac{3 a}{8} = \frac{1 - 4}{12}$

Step 4.4.5

Subtract $4$ from $1$ .

$- \frac{3 a}{8} = \frac{- 3}{12}$

Step 4.4.6

Cancel the common factor of $- 3$ and $12$ .

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Step 4.4.6.1

Factor $3$ out of $- 3$ .

$- \frac{3 a}{8} = \frac{3 (- 1)}{12}$

Step 4.4.6.2

Cancel the common factors.

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Step 4.4.6.2.1

Factor $3$ out of $12$ .

$- \frac{3 a}{8} = \frac{3 \cdot - 1}{3 \cdot 4}$

Step 4.4.6.2.2

Cancel the common factor.

$- \frac{3 a}{8} = \frac{3 \cdot - 1}{3 \cdot 4}$

Step 4.4.6.2.3

Rewrite the expression.

$- \frac{3 a}{8} = \frac{- 1}{4}$

Step 4.4.7

Move the negative in front of the fraction.

$- \frac{3 a}{8} = - \frac{1}{4}$

Step 4.5

Multiply both sides of the equation by $- \frac{8}{3}$ .

$- \frac{8}{3} (- \frac{3 a}{8}) = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6

Simplify both sides of the equation.

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Step 4.6.1

Simplify the left side.

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Step 4.6.1.1

Simplify $- \frac{8}{3} (- \frac{3 a}{8})$ .

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Step 4.6.1.1.1

Cancel the common factor of $8$ .

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Step 4.6.1.1.1.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$\frac{- 8}{3} (- \frac{3 a}{8}) = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.1.2

Move the leading negative in $- \frac{3 a}{8}$ into the numerator.

$\frac{- 8}{3} \cdot \frac{- 3 a}{8} = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.1.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{3} \cdot \frac{- 3 a}{8} = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.1.4

Cancel the common factor.

$\frac{8 \cdot - 1}{3} \cdot \frac{- 3 a}{8} = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{3} (- 3 a) = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.2

Cancel the common factor of $3$ .

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Step 4.6.1.1.2.1

Factor $3$ out of $- 3 a$ .

$\frac{- 1}{3} (3 (- a)) = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{3} (3 (- a)) = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.2.3

Rewrite the expression.

$- - a = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.3

Multiply.

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Step 4.6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 a = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.1.1.3.2

Multiply $a$ by $1$ .

$a = - \frac{8}{3} (- \frac{1}{4})$

Step 4.6.2

Simplify the right side.

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Step 4.6.2.1

Simplify $- \frac{8}{3} (- \frac{1}{4})$ .

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Step 4.6.2.1.1

Cancel the common factor of $4$ .

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Step 4.6.2.1.1.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$a = \frac{- 8}{3} (- \frac{1}{4})$

Step 4.6.2.1.1.2

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$a = \frac{- 8}{3} \cdot \frac{- 1}{4}$

Step 4.6.2.1.1.3

Factor $4$ out of $- 8$ .

$a = \frac{4 (- 2)}{3} \cdot \frac{- 1}{4}$

Step 4.6.2.1.1.4

Cancel the common factor.

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

Step 4.6.2.1.1.5

Rewrite the expression.

$a = \frac{- 2}{3} \cdot - 1$

Step 4.6.2.1.2

Combine $\frac{- 2}{3}$ and $- 1$ .

$a = \frac{- 2 \cdot - 1}{3}$

Step 4.6.2.1.3

Multiply $- 2$ by $- 1$ .

$a = \frac{2}{3}$

Step 5

Solve $\frac{1}{8} a + \frac{1}{3} = - (\frac{1}{2} a + \frac{1}{12})$ for $a$ .

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Step 5.1

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

$\frac{a}{8} + \frac{1}{3} = - (\frac{1}{2} a + \frac{1}{12})$

Step 5.2

Simplify $- (\frac{1}{2} a + \frac{1}{12})$ .

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Step 5.2.1

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

$\frac{a}{8} + \frac{1}{3} = - (\frac{a}{2} + \frac{1}{12})$

Step 5.2.2

Apply the distributive property.

$\frac{a}{8} + \frac{1}{3} = - \frac{a}{2} - \frac{1}{12}$

Step 5.3

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

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Step 5.3.1

Add $\frac{a}{2}$ to both sides of the equation.

$\frac{a}{8} + \frac{1}{3} + \frac{a}{2} = - \frac{1}{12}$

Step 5.3.2

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

$\frac{a}{8} + \frac{a}{2} \cdot \frac{4}{4} + \frac{1}{3} = - \frac{1}{12}$

Step 5.3.3

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

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Step 5.3.3.1

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

$\frac{a}{8} + \frac{a \cdot 4}{2 \cdot 4} + \frac{1}{3} = - \frac{1}{12}$

Step 5.3.3.2

Multiply $2$ by $4$ .

$\frac{a}{8} + \frac{a \cdot 4}{8} + \frac{1}{3} = - \frac{1}{12}$

Step 5.3.4

Combine the numerators over the common denominator.

$\frac{a + a \cdot 4}{8} + \frac{1}{3} = - \frac{1}{12}$

Step 5.3.5

Add $a$ and $a \cdot 4$ .

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Step 5.3.5.1

Reorder $a$ and $4$ .

$\frac{a + 4 \cdot a}{8} + \frac{1}{3} = - \frac{1}{12}$

Step 5.3.5.2

Add $a$ and $4 \cdot a$ .

$\frac{5 a}{8} + \frac{1}{3} = - \frac{1}{12}$

Step 5.4

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

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Step 5.4.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{5 a}{8} = - \frac{1}{12} - \frac{1}{3}$

Step 5.4.2

To write $- \frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{5 a}{8} = - \frac{1}{12} - \frac{1}{3} \cdot \frac{4}{4}$

Step 5.4.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.1

Multiply $\frac{1}{3}$ by $\frac{4}{4}$ .

$\frac{5 a}{8} = - \frac{1}{12} - \frac{4}{3 \cdot 4}$

Step 5.4.3.2

Multiply $3$ by $4$ .

$\frac{5 a}{8} = - \frac{1}{12} - \frac{4}{12}$

Step 5.4.4

Combine the numerators over the common denominator.

$\frac{5 a}{8} = \frac{- 1 - 4}{12}$

Step 5.4.5

Subtract $4$ from $- 1$ .

$\frac{5 a}{8} = \frac{- 5}{12}$

Step 5.4.6

Move the negative in front of the fraction.

$\frac{5 a}{8} = - \frac{5}{12}$

Step 5.5

Multiply both sides of the equation by $\frac{8}{5}$ .

$\frac{8}{5} \cdot \frac{5 a}{8} = \frac{8}{5} (- \frac{5}{12})$

Step 5.6

Simplify both sides of the equation.

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Step 5.6.1

Simplify the left side.

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Step 5.6.1.1

Simplify $\frac{8}{5} \cdot \frac{5 a}{8}$ .

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Step 5.6.1.1.1

Cancel the common factor of $8$ .

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Step 5.6.1.1.1.1

Cancel the common factor.

$\frac{8}{5} \cdot \frac{5 a}{8} = \frac{8}{5} (- \frac{5}{12})$

Step 5.6.1.1.1.2

Rewrite the expression.

$\frac{1}{5} (5 a) = \frac{8}{5} (- \frac{5}{12})$

Step 5.6.1.1.2

Cancel the common factor of $5$ .

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Step 5.6.1.1.2.1

Factor $5$ out of $5 a$ .

$\frac{1}{5} (5 (a)) = \frac{8}{5} (- \frac{5}{12})$

Step 5.6.1.1.2.2

Cancel the common factor.

$\frac{1}{5} (5 a) = \frac{8}{5} (- \frac{5}{12})$

Step 5.6.1.1.2.3

Rewrite the expression.

$a = \frac{8}{5} (- \frac{5}{12})$

Step 5.6.2

Simplify the right side.

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Step 5.6.2.1

Simplify $\frac{8}{5} (- \frac{5}{12})$ .

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Step 5.6.2.1.1

Cancel the common factor of $4$ .

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Step 5.6.2.1.1.1

Move the leading negative in $- \frac{5}{12}$ into the numerator.

$a = \frac{8}{5} \cdot \frac{- 5}{12}$

Step 5.6.2.1.1.2

Factor $4$ out of $8$ .

$a = \frac{4 (2)}{5} \cdot \frac{- 5}{12}$

Step 5.6.2.1.1.3

Factor $4$ out of $12$ .

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

Step 5.6.2.1.1.4

Cancel the common factor.

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

Step 5.6.2.1.1.5

Rewrite the expression.

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

Step 5.6.2.1.2

Cancel the common factor of $5$ .

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Step 5.6.2.1.2.1

Factor $5$ out of $- 5$ .

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

Step 5.6.2.1.2.2

Cancel the common factor.

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

Step 5.6.2.1.2.3

Rewrite the expression.

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

Step 5.6.2.1.3

Combine $2$ and $\frac{- 1}{3}$ .

$a = \frac{2 \cdot - 1}{3}$

Step 5.6.2.1.4

Simplify the expression.

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Step 5.6.2.1.4.1

Multiply $2$ by $- 1$ .

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

Step 5.6.2.1.4.2

Move the negative in front of the fraction.

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

Step 6

List all of the solutions.

$a = \frac{2}{3}, - \frac{2}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{2}{3}, - \frac{2}{3}$

Decimal Form:

$a = 0.\overline{6}, - 0.\overline{6}$