Basic Math Examples

$| z + (\frac{1}{8}) | = | z - \frac{5}{8} |$

Step 1

Rewrite the equation as $| z - \frac{5}{8} | = | z + \frac{1}{8} |$ .

$| z - \frac{5}{8} | = | z + \frac{1}{8} |$

Step 2

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

$z - \frac{5}{8} = z + \frac{1}{8}$

$z - \frac{5}{8} = - (z + \frac{1}{8})$

$- (z - \frac{5}{8}) = z + \frac{1}{8}$

$- (z - \frac{5}{8}) = - (z + \frac{1}{8})$

Step 3

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

$z - \frac{5}{8} = z + \frac{1}{8}$

$z - \frac{5}{8} = - (z + \frac{1}{8})$

Step 4

Solve $z - \frac{5}{8} = z + \frac{1}{8}$ for $z$ .

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Step 4.1

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

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Step 4.1.1

Subtract $z$ from both sides of the equation.

$z - \frac{5}{8} - z = \frac{1}{8}$

Step 4.1.2

Combine the opposite terms in $z - \frac{5}{8} - z$ .

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Step 4.1.2.1

Subtract $z$ from $z$ .

$0 - \frac{5}{8} = \frac{1}{8}$

Step 4.1.2.2

Subtract $\frac{5}{8}$ from $0$ .

$- \frac{5}{8} = \frac{1}{8}$

Step 4.2

Since $- \frac{5}{8} \neq \frac{1}{8}$ , there are no solutions.

No solution

Step 5

Solve $z - \frac{5}{8} = - (z + \frac{1}{8})$ for $z$ .

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Step 5.1

Simplify $- (z + \frac{1}{8})$ .

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Step 5.1.1

Rewrite.

$z - \frac{5}{8} = 0 + 0 - (z + \frac{1}{8})$

Step 5.1.2

Simplify by adding zeros.

$z - \frac{5}{8} = - (z + \frac{1}{8})$

Step 5.1.3

Apply the distributive property.

$z - \frac{5}{8} = - z - \frac{1}{8}$

Step 5.2

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

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Step 5.2.1

Add $z$ to both sides of the equation.

$z - \frac{5}{8} + z = - \frac{1}{8}$

Step 5.2.2

Add $z$ and $z$ .

$2 z - \frac{5}{8} = - \frac{1}{8}$

Step 5.3

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

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Step 5.3.1

Add $\frac{5}{8}$ to both sides of the equation.

$2 z = - \frac{1}{8} + \frac{5}{8}$

Step 5.3.2

Combine the numerators over the common denominator.

$2 z = \frac{- 1 + 5}{8}$

Step 5.3.3

Add $- 1$ and $5$ .

$2 z = \frac{4}{8}$

Step 5.3.4

Cancel the common factor of $4$ and $8$ .

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Step 5.3.4.1

Factor $4$ out of $4$ .

$2 z = \frac{4 (1)}{8}$

Step 5.3.4.2

Cancel the common factors.

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Step 5.3.4.2.1

Factor $4$ out of $8$ .

$2 z = \frac{4 \cdot 1}{4 \cdot 2}$

Step 5.3.4.2.2

Cancel the common factor.

$2 z = \frac{4 \cdot 1}{4 \cdot 2}$

Step 5.3.4.2.3

Rewrite the expression.

$2 z = \frac{1}{2}$

Step 5.4

Divide each term in $2 z = \frac{1}{2}$ by $2$ and simplify.

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Step 5.4.1

Divide each term in $2 z = \frac{1}{2}$ by $2$ .

$\frac{2 z}{2} = \frac{\frac{1}{2}}{2}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{\frac{1}{2}}{2}$

Step 5.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{\frac{1}{2}}{2}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$z = \frac{1}{2} \cdot \frac{1}{2}$

Step 5.4.3.2

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 5.4.3.2.1

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

$z = \frac{1}{2 \cdot 2}$

Step 5.4.3.2.2

Multiply $2$ by $2$ .

$z = \frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$z = \frac{1}{4}$

Decimal Form:

$z = 0.25$