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Solution - Absolute value equations

Exact form:

z = - 3, - \frac{5}{3}

z=-3 , -\frac{5}{3}

Mixed number form:

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

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

Decimal form:

z = - 3, - 1.667

z=-3 , -1.667

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| z + 1 | = 2 | z + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| z + 1 \| = 2 \| z + 2 \|$
$x = + y$	$(z + 1) = 2 (z + 2)$
$x = - y$	$(z + 1) = 2 (- (z + 2))$
$+ x = y$	$(z + 1) = 2 (z + 2)$
$- x = y$	$- (z + 1) = 2 (z + 2)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| z + 1 \| = 2 \| z + 2 \|$
$x = + y$ , $+ x = y$	$(z + 1) = 2 (z + 2)$
$x = - y$ , $- x = y$	$(z + 1) = 2 (- (z + 2))$

2. Solve the two equations for z

12 additional steps

$(z + 1) = 2 \cdot (z + 2)$

Expand the parentheses:

$(z + 1) = 2 z + 2 \cdot 2$

Simplify the arithmetic:

$(z + 1) = 2 z + 4$

Subtract from both sides:

$(z + 1) - 2 z = (2 z + 4) - 2 z$

Group like terms:

$(z - 2 z) + 1 = (2 z + 4) - 2 z$

Simplify the arithmetic:

$- z + 1 = (2 z + 4) - 2 z$

Group like terms:

$- z + 1 = (2 z - 2 z) + 4$

Simplify the arithmetic:

$- z + 1 = 4$

Subtract from both sides:

$(- z + 1) - 1 = 4 - 1$

Simplify the arithmetic:

$- z = 4 - 1$

Simplify the arithmetic:

$- z = 3$

Multiply both sides by :

$- z \cdot - 1 = 3 \cdot - 1$

Remove the one(s):

$z = 3 \cdot - 1$

Simplify the arithmetic:

$z = - 3$

14 additional steps

$(z + 1) = 2 \cdot (- (z + 2))$

Expand the parentheses:

$(z + 1) = 2 \cdot (- z - 2)$

$(z + 1) = 2 \cdot - z + 2 \cdot - 2$

Group like terms:

$(z + 1) = (2 \cdot - 1) z + 2 \cdot - 2$

Multiply the coefficients:

$(z + 1) = - 2 z + 2 \cdot - 2$

Simplify the arithmetic:

$(z + 1) = - 2 z - 4$

Add to both sides:

$(z + 1) + 2 z = (- 2 z - 4) + 2 z$

Group like terms:

$(z + 2 z) + 1 = (- 2 z - 4) + 2 z$

Simplify the arithmetic:

$3 z + 1 = (- 2 z - 4) + 2 z$

Group like terms:

$3 z + 1 = (- 2 z + 2 z) - 4$

Simplify the arithmetic:

$3 z + 1 = - 4$

Subtract from both sides:

$(3 z + 1) - 1 = - 4 - 1$

Simplify the arithmetic:

$3 z = - 4 - 1$

Simplify the arithmetic:

$3 z = - 5$

Divide both sides by :

$\frac{(3 z)}{3} = \frac{- 5}{3}$

Simplify the fraction:

$z = \frac{- 5}{3}$

3. List the solutions

$z = - 3, - \frac{5}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | z + 1 |$
$y = 2 | z + 2 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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