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Solution - Absolute value equations

Exact form:

z = - 6, - 6

z=-6 , -6

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
$0 | z - 4 | = | z + 6 |$
without the absolute value bars:

$\| x \| = \| y \|$	$0 \| z - 4 \| = \| z + 6 \|$
$x = + y$	$0 (z - 4) = (z + 6)$
$x = - y$	$0 (z - 4) = - (z + 6)$
$+ x = y$	$0 (z - 4) = (z + 6)$
$- x = y$	$0 (- (z - 4)) = (z + 6)$

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:

$\| x \| = \| y \|$	$0 \| z - 4 \| = \| z + 6 \|$
$x = + y$ , $+ x = y$	$0 (z - 4) = (z + 6)$
$x = - y$ , $- x = y$	$0 (z - 4) = - (z + 6)$

2. Solve the two equations for z

4 additional steps

$0 \cdot (z - 4) = (z + 6)$

NT_MSLUS_MAINSTEP_MULTIPLY_BY_ZERO:

$0 = (z + 6)$

Swap sides:

$(z + 6) = 0$

Subtract from both sides:

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

Simplify the arithmetic:

$z = 0 - 6$

Simplify the arithmetic:

$z = - 6$

8 additional steps

$0 \cdot (z - 4) = - (z + 6)$

NT_MSLUS_MAINSTEP_MULTIPLY_BY_ZERO:

$0 = - (z + 6)$

Expand the parentheses:

$0 = - z - 6$

Swap sides:

$- z - 6 = 0$

Add to both sides:

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

Simplify the arithmetic:

$- z = 0 + 6$

Simplify the arithmetic:

$- z = 6$

Multiply both sides by :

$- z \cdot - 1 = 6 \cdot - 1$

Remove the one(s):

$z = 6 \cdot - 1$

Simplify the arithmetic:

$z = - 6$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = 0 | z - 4 |$
$y = | z + 6 |$
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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