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Solution - Absolute value equations

Exact form:

z = - \frac{7}{2}, - \frac{5}{4}

z=-\frac{7}{2} , -\frac{5}{4}

Mixed number form:

z = - 3 \frac{1}{2}, - 1 \frac{1}{4}

z=-3\frac{1}{2} , -1\frac{1}{4}

Decimal form:

z = - 3.5, - 1.25

z=-3.5 , -1.25

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
$| z - 1 | = 3 | z + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| z - 1 \| = 3 \| z + 2 \|$
$x = + y$	$(z - 1) = 3 (z + 2)$
$x = - y$	$(z - 1) = 3 (- (z + 2))$
$+ x = y$	$(z - 1) = 3 (z + 2)$
$- x = y$	$- (z - 1) = 3 (z + 2)$

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:

$\| x \| = \| y \|$	$\| z - 1 \| = 3 \| z + 2 \|$
$x = + y$ , $+ x = y$	$(z - 1) = 3 (z + 2)$
$x = - y$ , $- x = y$	$(z - 1) = 3 (- (z + 2))$

2. Solve the two equations for z

13 additional steps

$(z - 1) = 3 \cdot (z + 2)$

Expand the parentheses:

$(z - 1) = 3 z + 3 \cdot 2$

Simplify the arithmetic:

$(z - 1) = 3 z + 6$

Subtract from both sides:

$(z - 1) - 3 z = (3 z + 6) - 3 z$

Group like terms:

$(z - 3 z) - 1 = (3 z + 6) - 3 z$

Simplify the arithmetic:

$- 2 z - 1 = (3 z + 6) - 3 z$

Group like terms:

$- 2 z - 1 = (3 z - 3 z) + 6$

Simplify the arithmetic:

$- 2 z - 1 = 6$

Add to both sides:

$(- 2 z - 1) + 1 = 6 + 1$

Simplify the arithmetic:

$- 2 z = 6 + 1$

Simplify the arithmetic:

$- 2 z = 7$

Divide both sides by :

$\frac{(- 2 z)}{- 2} = \frac{7}{- 2}$

Cancel out the negatives:

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

Simplify the fraction:

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

Move the negative sign from the denominator to the numerator:

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

14 additional steps

$(z - 1) = 3 \cdot (- (z + 2))$

Expand the parentheses:

$(z - 1) = 3 \cdot (- z - 2)$

$(z - 1) = 3 \cdot - z + 3 \cdot - 2$

Group like terms:

$(z - 1) = (3 \cdot - 1) z + 3 \cdot - 2$

Multiply the coefficients:

$(z - 1) = - 3 z + 3 \cdot - 2$

Simplify the arithmetic:

$(z - 1) = - 3 z - 6$

Add to both sides:

$(z - 1) + 3 z = (- 3 z - 6) + 3 z$

Group like terms:

$(z + 3 z) - 1 = (- 3 z - 6) + 3 z$

Simplify the arithmetic:

$4 z - 1 = (- 3 z - 6) + 3 z$

Group like terms:

$4 z - 1 = (- 3 z + 3 z) - 6$

Simplify the arithmetic:

$4 z - 1 = - 6$

Add to both sides:

$(4 z - 1) + 1 = - 6 + 1$

Simplify the arithmetic:

$4 z = - 6 + 1$

Simplify the arithmetic:

$4 z = - 5$

Divide both sides by :

$\frac{(4 z)}{4} = \frac{- 5}{4}$

Simplify the fraction:

$z = \frac{- 5}{4}$

3. List the solutions

$z = - \frac{7}{2}, - \frac{5}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | z - 1 |$
$y = 3 | z + 2 |$
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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