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Solution - Absolute value equations

Exact form:

z = - 6, - 2

z=-6 , -2

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
$| 2 z + 6 | = | z |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 z + 6 \| = \| z \|$
$x = + y$	$(2 z + 6) = (z)$
$x = - y$	$(2 z + 6) = - (z)$
$+ x = y$	$(2 z + 6) = (z)$
$- x = y$	$- (2 z + 6) = (z)$

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:

$\| x \| = \| y \|$	$\| 2 z + 6 \| = \| z \|$
$x = + y$ , $+ x = y$	$(2 z + 6) = (z)$
$x = - y$ , $- x = y$	$(2 z + 6) = - (z)$

2. Solve the two equations for z

6 additional steps

$(2 z + 6) = z$

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$z + 6 = z - z$

Simplify the arithmetic:

$z + 6 = 0$

Subtract from both sides:

$(z + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$z = 0 - 6$

Simplify the arithmetic:

$z = - 6$

10 additional steps

$(2 z + 6) = - z$

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$3 z + 6 = - z + z$

Simplify the arithmetic:

$3 z + 6 = 0$

Subtract from both sides:

$(3 z + 6) - 6 = 0 - 6$

Simplify the arithmetic:

$3 z = 0 - 6$

Simplify the arithmetic:

$3 z = - 6$

Divide both sides by :

$\frac{(3 z)}{3} = \frac{- 6}{3}$

Simplify the fraction:

$z = \frac{- 6}{3}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$z = - 2$

3. List the solutions

$z = - 6, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 z + 6 |$
$y = | z |$
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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