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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

z = 1, - 0.5

z=1 , -0.5

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

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

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:

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

2. Solve the two equations for z

11 additional steps

$(z + 2) = 3 z$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$- 2 z + 2 = 0$

Subtract from both sides:

$(- 2 z + 2) - 2 = 0 - 2$

Simplify the arithmetic:

$- 2 z = 0 - 2$

Simplify the arithmetic:

$- 2 z = - 2$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

Simplify the fraction:

$z = 1$

12 additional steps

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

Group like terms:

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

Multiply the coefficients:

$(z + 2) = - 3 z$

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Simplify the arithmetic:

$4 z + 2 = 0$

Subtract from both sides:

$(4 z + 2) - 2 = 0 - 2$

Simplify the arithmetic:

$4 z = 0 - 2$

Simplify the arithmetic:

$4 z = - 2$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

$z = 1, - \frac{1}{2}$
(2 solution(s))

4. Graph

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