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Solution - Absolute value equations

Exact form:

z = 1, - 3

z=1 , -3

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
$| 8 z + 20 | = | 6 z + 22 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 z + 20 \| = \| 6 z + 22 \|$
$x = + y$	$(8 z + 20) = (6 z + 22)$
$x = - y$	$(8 z + 20) = - (6 z + 22)$
$+ x = y$	$(8 z + 20) = (6 z + 22)$
$- x = y$	$- (8 z + 20) = (6 z + 22)$

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:

$\| x \| = \| y \|$	$\| 8 z + 20 \| = \| 6 z + 22 \|$
$x = + y$ , $+ x = y$	$(8 z + 20) = (6 z + 22)$
$x = - y$ , $- x = y$	$(8 z + 20) = - (6 z + 22)$

2. Solve the two equations for z

10 additional steps

$(8 z + 20) = (6 z + 22)$

Subtract from both sides:

$(8 z + 20) - 6 z = (6 z + 22) - 6 z$

Group like terms:

$(8 z - 6 z) + 20 = (6 z + 22) - 6 z$

Simplify the arithmetic:

$2 z + 20 = (6 z + 22) - 6 z$

Group like terms:

$2 z + 20 = (6 z - 6 z) + 22$

Simplify the arithmetic:

$2 z + 20 = 22$

Subtract from both sides:

$(2 z + 20) - 20 = 22 - 20$

Simplify the arithmetic:

$2 z = 22 - 20$

Simplify the arithmetic:

$2 z = 2$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$z = 1$

12 additional steps

$(8 z + 20) = - (6 z + 22)$

Expand the parentheses:

$(8 z + 20) = - 6 z - 22$

Add to both sides:

$(8 z + 20) + 6 z = (- 6 z - 22) + 6 z$

Group like terms:

$(8 z + 6 z) + 20 = (- 6 z - 22) + 6 z$

Simplify the arithmetic:

$14 z + 20 = (- 6 z - 22) + 6 z$

Group like terms:

$14 z + 20 = (- 6 z + 6 z) - 22$

Simplify the arithmetic:

$14 z + 20 = - 22$

Subtract from both sides:

$(14 z + 20) - 20 = - 22 - 20$

Simplify the arithmetic:

$14 z = - 22 - 20$

Simplify the arithmetic:

$14 z = - 42$

Divide both sides by :

$\frac{(14 z)}{14} = \frac{- 42}{14}$

Simplify the fraction:

$z = \frac{- 42}{14}$

Find the greatest common factor of the numerator and denominator:

$z = \frac{(- 3 \cdot 14)}{(1 \cdot 14)}$

Factor out and cancel the greatest common factor:

$z = - 3$

3. List the solutions

$z = 1, - 3$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 z + 20 |$
$y = | 6 z + 22 |$
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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