Algebra Examples

$2 | z + 1 | - 3 = z - 2$

Step 1

Move all terms not containing $| z + 1 |$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$2 | z + 1 | = z - 2 + 3$

Step 1.2

Add $- 2$ and $3$ .

$2 | z + 1 | = z + 1$

Step 2

Divide each term in $2 | z + 1 | = z + 1$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 | z + 1 | = z + 1$ by $2$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Divide $| z + 1 |$ by $1$ .

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

Step 3

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

$z + 1 = \pm (\frac{z}{2} + \frac{1}{2})$

Step 4

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

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Subtract $z$ from $z \cdot 2$ .

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Step 4.2.5.1

Reorder $z$ and $2$ .

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

Step 4.2.5.2

Subtract $z$ from $2 \cdot z$ .

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

Step 4.3

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

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Step 4.3.1

Subtract $1$ from both sides of the equation.

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Combine the numerators over the common denominator.

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

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $- 1$ by $2$ .

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

Step 4.3.5.2

Subtract $2$ from $1$ .

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

Step 4.3.6

Move the negative in front of the fraction.

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

Step 4.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$z = - 1$

Step 4.5

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

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

Step 4.6

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

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Step 4.6.1

Rewrite.

$z + 1 = 0 + 0 - (\frac{z}{2} + \frac{1}{2})$

Step 4.6.2

Simplify by adding zeros.

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

Step 4.6.3

Apply the distributive property.

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Combine the numerators over the common denominator.

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

Step 4.7.5

Add $z \cdot 2$ and $z$ .

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Step 4.7.5.1

Reorder $z$ and $2$ .

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

Step 4.7.5.2

Add $2 \cdot z$ and $z$ .

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

$\frac{3 z}{2} + 1 = - \frac{1}{2}$

Step 4.8

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

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Step 4.8.1

Subtract $1$ from both sides of the equation.

$\frac{3 z}{2} = - \frac{1}{2} - 1$

Step 4.8.2

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

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

Step 4.8.3

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

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

Step 4.8.4

Combine the numerators over the common denominator.

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

Step 4.8.5

Simplify the numerator.

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Step 4.8.5.1

Multiply $- 1$ by $2$ .

$\frac{3 z}{2} = \frac{- 1 - 2}{2}$

Step 4.8.5.2

Subtract $2$ from $- 1$ .

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

Step 4.8.6

Move the negative in front of the fraction.

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

Step 4.9

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$3 z = - 3$

Step 4.10

Divide each term in $3 z = - 3$ by $3$ and simplify.

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Step 4.10.1

Divide each term in $3 z = - 3$ by $3$ .

$\frac{3 z}{3} = \frac{- 3}{3}$

Step 4.10.2

Simplify the left side.

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Step 4.10.2.1

Cancel the common factor of $3$ .

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Step 4.10.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{- 3}{3}$

Step 4.10.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 3}{3}$

Step 4.10.3

Simplify the right side.

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Step 4.10.3.1

Divide $- 3$ by $3$ .

$z = - 1$