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Solution - Absolute value equations

Exact form:

z = - 1, \frac{1}{3}

z=-1 , \frac{1}{3}

Decimal form:

z = - 1, 0.333

z=-1 , 0.333

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
$| z - 1 | = 2 | z |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| z - 1 \| = 2 \| z \|$
$x = + y$	$(z - 1) = 2 (z)$
$x = - y$	$(z - 1) = 2 (- (z))$
$+ x = y$	$(z - 1) = 2 (z)$
$- x = y$	$- (z - 1) = 2 (z)$

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:

$\| x \| = \| y \|$	$\| z - 1 \| = 2 \| z \|$
$x = + y$ , $+ x = y$	$(z - 1) = 2 (z)$
$x = - y$ , $- x = y$	$(z - 1) = 2 (- (z))$

2. Solve the two equations for z

9 additional steps

$(z - 1) = 2 z$

Subtract from both sides:

$(z - 1) - 2 z = (2 z) - 2 z$

Group like terms:

$(z - 2 z) - 1 = (2 z) - 2 z$

Simplify the arithmetic:

$- z - 1 = (2 z) - 2 z$

Simplify the arithmetic:

$- z - 1 = 0$

Add to both sides:

$(- z - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$- z = 0 + 1$

Simplify the arithmetic:

$- z = 1$

Multiply both sides by :

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

Remove the one(s):

$z = 1 \cdot - 1$

Remove the one(s):

$z = - 1$

10 additional steps

$(z - 1) = 2 \cdot - z$

Group like terms:

$(z - 1) = (2 \cdot - 1) z$

Multiply the coefficients:

$(z - 1) = - 2 z$

Add to both sides:

$(z - 1) + 2 z = (- 2 z) + 2 z$

Group like terms:

$(z + 2 z) - 1 = (- 2 z) + 2 z$

Simplify the arithmetic:

$3 z - 1 = (- 2 z) + 2 z$

Simplify the arithmetic:

$3 z - 1 = 0$

Add to both sides:

$(3 z - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$3 z = 0 + 1$

Simplify the arithmetic:

$3 z = 1$

Divide both sides by :

$\frac{(3 z)}{3} = \frac{1}{3}$

Simplify the fraction:

$z = \frac{1}{3}$

3. List the solutions

$z = - 1, \frac{1}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | z - 1 |$
$y = 2 | z |$
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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