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Solution - Absolute value equations

Exact form:

z = 5, 1

z=5 , 1

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
$| 3 z - 5 | = | 2 z |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 z - 5 \| = \| 2 z \|$
$x = + y$	$(3 z - 5) = (2 z)$
$x = - y$	$(3 z - 5) = - (2 z)$
$+ x = y$	$(3 z - 5) = (2 z)$
$- x = y$	$- (3 z - 5) = (2 z)$

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:

$\| x \| = \| y \|$	$\| 3 z - 5 \| = \| 2 z \|$
$x = + y$ , $+ x = y$	$(3 z - 5) = (2 z)$
$x = - y$ , $- x = y$	$(3 z - 5) = - (2 z)$

2. Solve the two equations for z

6 additional steps

$(3 z - 5) = 2 z$

Subtract from both sides:

$(3 z - 5) - 2 z = (2 z) - 2 z$

Group like terms:

$(3 z - 2 z) - 5 = (2 z) - 2 z$

Simplify the arithmetic:

$z - 5 = (2 z) - 2 z$

Simplify the arithmetic:

$z - 5 = 0$

Add to both sides:

$(z - 5) + 5 = 0 + 5$

Simplify the arithmetic:

$z = 0 + 5$

Simplify the arithmetic:

$z = 5$

8 additional steps

$(3 z - 5) = - 2 z$

Add to both sides:

$(3 z - 5) + 5 = (- 2 z) + 5$

Simplify the arithmetic:

$3 z = (- 2 z) + 5$

Add to both sides:

$(3 z) + 2 z = ((- 2 z) + 5) + 2 z$

Simplify the arithmetic:

$5 z = ((- 2 z) + 5) + 2 z$

Group like terms:

$5 z = (- 2 z + 2 z) + 5$

Simplify the arithmetic:

$5 z = 5$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$z = 1$

3. List the solutions

$z = 5, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 z - 5 |$
$y = | 2 z |$
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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