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Solution - Absolute value equations

Exact form:

z = 4, \frac{4}{9}

z=4 , \frac{4}{9}

Decimal form:

z = 4, 0.444

z=4 , 0.444

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
$| 5 z - 4 | = | 4 z |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 z - 4 \| = \| 4 z \|$
$x = + y$	$(5 z - 4) = (4 z)$
$x = - y$	$(5 z - 4) = - (4 z)$
$+ x = y$	$(5 z - 4) = (4 z)$
$- x = y$	$- (5 z - 4) = (4 z)$

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:

$\| x \| = \| y \|$	$\| 5 z - 4 \| = \| 4 z \|$
$x = + y$ , $+ x = y$	$(5 z - 4) = (4 z)$
$x = - y$ , $- x = y$	$(5 z - 4) = - (4 z)$

2. Solve the two equations for z

6 additional steps

$(5 z - 4) = 4 z$

Subtract from both sides:

$(5 z - 4) - 4 z = (4 z) - 4 z$

Group like terms:

$(5 z - 4 z) - 4 = (4 z) - 4 z$

Simplify the arithmetic:

$z - 4 = (4 z) - 4 z$

Simplify the arithmetic:

$z - 4 = 0$

Add to both sides:

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

Simplify the arithmetic:

$z = 0 + 4$

Simplify the arithmetic:

$z = 4$

7 additional steps

$(5 z - 4) = - 4 z$

Add to both sides:

$(5 z - 4) + 4 = (- 4 z) + 4$

Simplify the arithmetic:

$5 z = (- 4 z) + 4$

Add to both sides:

$(5 z) + 4 z = ((- 4 z) + 4) + 4 z$

Simplify the arithmetic:

$9 z = ((- 4 z) + 4) + 4 z$

Group like terms:

$9 z = (- 4 z + 4 z) + 4$

Simplify the arithmetic:

$9 z = 4$

Divide both sides by :

$\frac{(9 z)}{9} = \frac{4}{9}$

Simplify the fraction:

$z = \frac{4}{9}$

3. List the solutions

$z = 4, \frac{4}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 z - 4 |$
$y = | 4 z |$
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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