Algebra Examples

$| \frac{z}{2} + 4 | = | 4 z - 6 |$

Step 1

Rewrite the equation as $| 4 z - 6 | = | \frac{z}{2} + 4 |$ .

$| 4 z - 6 | = | \frac{z}{2} + 4 |$

Step 2

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

$4 z - 6 = \frac{z}{2} + 4$

$4 z - 6 = - (\frac{z}{2} + 4)$

$- (4 z - 6) = \frac{z}{2} + 4$

$- (4 z - 6) = - (\frac{z}{2} + 4)$

Step 3

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

$4 z - 6 = \frac{z}{2} + 4$

$4 z - 6 = - (\frac{z}{2} + 4)$

Step 4

Solve $4 z - 6 = \frac{z}{2} + 4$ 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 $\frac{z}{2}$ from both sides of the equation.

$4 z - 6 - \frac{z}{2} = 4$

Step 4.1.2

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

$4 z \cdot \frac{2}{2} - \frac{z}{2} - 6 = 4$

Step 4.1.3

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

$\frac{4 z \cdot 2}{2} - \frac{z}{2} - 6 = 4$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{4 z \cdot 2 - z}{2} - 6 = 4$

Step 4.1.5

Simplify each term.

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Step 4.1.5.1

Simplify the numerator.

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Step 4.1.5.1.1

Factor $z$ out of $4 z \cdot 2 - z$ .

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Step 4.1.5.1.1.1

Factor $z$ out of $4 z \cdot 2$ .

$\frac{z (4 \cdot 2) - z}{2} - 6 = 4$

Step 4.1.5.1.1.2

Factor $z$ out of $- z$ .

$\frac{z (4 \cdot 2) + z \cdot - 1}{2} - 6 = 4$

Step 4.1.5.1.1.3

Factor $z$ out of $z (4 \cdot 2) + z \cdot - 1$ .

$\frac{z (4 \cdot 2 - 1)}{2} - 6 = 4$

Step 4.1.5.1.2

Multiply $4$ by $2$ .

$\frac{z (8 - 1)}{2} - 6 = 4$

Step 4.1.5.1.3

Subtract $1$ from $8$ .

$\frac{z \cdot 7}{2} - 6 = 4$

Step 4.1.5.2

Move $7$ to the left of $z$ .

$\frac{7 z}{2} - 6 = 4$

Step 4.2

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

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Step 4.2.1

Add $6$ to both sides of the equation.

$\frac{7 z}{2} = 4 + 6$

Step 4.2.2

Add $4$ and $6$ .

$\frac{7 z}{2} = 10$

Step 4.3

Multiply both sides of the equation by $\frac{2}{7}$ .

$\frac{2}{7} \cdot \frac{7 z}{2} = \frac{2}{7} \cdot 10$

Step 4.4

Simplify both sides of the equation.

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Step 4.4.1

Simplify the left side.

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Step 4.4.1.1

Simplify $\frac{2}{7} \cdot \frac{7 z}{2}$ .

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Step 4.4.1.1.1

Cancel the common factor of $2$ .

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Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{2}{7} \cdot \frac{7 z}{2} = \frac{2}{7} \cdot 10$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{7} (7 z) = \frac{2}{7} \cdot 10$

Step 4.4.1.1.2

Cancel the common factor of $7$ .

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Step 4.4.1.1.2.1

Factor $7$ out of $7 z$ .

$\frac{1}{7} (7 (z)) = \frac{2}{7} \cdot 10$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{7} (7 z) = \frac{2}{7} \cdot 10$

Step 4.4.1.1.2.3

Rewrite the expression.

$z = \frac{2}{7} \cdot 10$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Multiply $\frac{2}{7} \cdot 10$ .

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Step 4.4.2.1.1

Combine $\frac{2}{7}$ and $10$ .

$z = \frac{2 \cdot 10}{7}$

Step 4.4.2.1.2

Multiply $2$ by $10$ .

$z = \frac{20}{7}$

Step 5

Solve $4 z - 6 = - (\frac{z}{2} + 4)$ for $z$ .

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Step 5.1

Simplify $- (\frac{z}{2} + 4)$ .

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Step 5.1.1

Rewrite.

$4 z - 6 = 0 + 0 - (\frac{z}{2} + 4)$

Step 5.1.2

Simplify by adding zeros.

$4 z - 6 = - (\frac{z}{2} + 4)$

Step 5.1.3

Apply the distributive property.

$4 z - 6 = - \frac{z}{2} - 1 \cdot 4$

Step 5.1.4

Multiply $- 1$ by $4$ .

$4 z - 6 = - \frac{z}{2} - 4$

Step 5.2

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

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Step 5.2.1

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

$4 z - 6 + \frac{z}{2} = - 4$

Step 5.2.2

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

$4 z \cdot \frac{2}{2} + \frac{z}{2} - 6 = - 4$

Step 5.2.3

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

$\frac{4 z \cdot 2}{2} + \frac{z}{2} - 6 = - 4$

Step 5.2.4

Combine the numerators over the common denominator.

$\frac{4 z \cdot 2 + z}{2} - 6 = - 4$

Step 5.2.5

Simplify each term.

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Step 5.2.5.1

Simplify the numerator.

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Step 5.2.5.1.1

Factor $z$ out of $4 z \cdot 2 + z$ .

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Step 5.2.5.1.1.1

Factor $z$ out of $4 z \cdot 2$ .

$\frac{z (4 \cdot 2) + z}{2} - 6 = - 4$

Step 5.2.5.1.1.2

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

$\frac{z (4 \cdot 2) + z^{1}}{2} - 6 = - 4$

Step 5.2.5.1.1.3

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

$\frac{z (4 \cdot 2) + z \cdot 1}{2} - 6 = - 4$

Step 5.2.5.1.1.4

Factor $z$ out of $z (4 \cdot 2) + z \cdot 1$ .

$\frac{z (4 \cdot 2 + 1)}{2} - 6 = - 4$

Step 5.2.5.1.2

Multiply $4$ by $2$ .

$\frac{z (8 + 1)}{2} - 6 = - 4$

Step 5.2.5.1.3

Add $8$ and $1$ .

$\frac{z \cdot 9}{2} - 6 = - 4$

Step 5.2.5.2

Move $9$ to the left of $z$ .

$\frac{9 z}{2} - 6 = - 4$

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 $6$ to both sides of the equation.

$\frac{9 z}{2} = - 4 + 6$

Step 5.3.2

Add $- 4$ and $6$ .

$\frac{9 z}{2} = 2$

Step 5.4

Multiply both sides of the equation by $\frac{2}{9}$ .

$\frac{2}{9} \cdot \frac{9 z}{2} = \frac{2}{9} \cdot 2$

Step 5.5

Simplify both sides of the equation.

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Step 5.5.1

Simplify the left side.

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Step 5.5.1.1

Simplify $\frac{2}{9} \cdot \frac{9 z}{2}$ .

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Step 5.5.1.1.1

Cancel the common factor of $2$ .

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Step 5.5.1.1.1.1

Cancel the common factor.

$\frac{2}{9} \cdot \frac{9 z}{2} = \frac{2}{9} \cdot 2$

Step 5.5.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 z) = \frac{2}{9} \cdot 2$

Step 5.5.1.1.2

Cancel the common factor of $9$ .

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Step 5.5.1.1.2.1

Factor $9$ out of $9 z$ .

$\frac{1}{9} (9 (z)) = \frac{2}{9} \cdot 2$

Step 5.5.1.1.2.2

Cancel the common factor.

$\frac{1}{9} (9 z) = \frac{2}{9} \cdot 2$

Step 5.5.1.1.2.3

Rewrite the expression.

$z = \frac{2}{9} \cdot 2$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Multiply $\frac{2}{9} \cdot 2$ .

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Step 5.5.2.1.1

Combine $\frac{2}{9}$ and $2$ .

$z = \frac{2 \cdot 2}{9}$

Step 5.5.2.1.2

Multiply $2$ by $2$ .

$z = \frac{4}{9}$

Step 6

List all of the solutions.

$z = \frac{20}{7}, \frac{4}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{20}{7}, \frac{4}{9}$

Decimal Form:

$z = 2.\overline{857142}, 0.\overline{4}$

Mixed Number Form:

$z = 2 \frac{6}{7}, \frac{4}{9}$