Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

z = 8, \frac{8}{3}

z=8 , \frac{8}{3}

Mixed number form:

z = 8, 2 \frac{2}{3}

z=8 , 2\frac{2}{3}

Decimal form:

z = 8, 2.667

z=8 , 2.667

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
$| z | = 2 | z - 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| z \| = 2 \| z - 4 \|$
$x = + y$	$(z) = 2 (z - 4)$
$x = - y$	$(z) = 2 (- (z - 4))$
$+ x = y$	$(z) = 2 (z - 4)$
$- x = y$	$- (z) = 2 (z - 4)$

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:

$\| x \| = \| y \|$	$\| z \| = 2 \| z - 4 \|$
$x = + y$ , $+ x = y$	$(z) = 2 (z - 4)$
$x = - y$ , $- x = y$	$(z) = 2 (- (z - 4))$

2. Solve the two equations for z

8 additional steps

$z = 2 \cdot (z - 4)$

Expand the parentheses:

$z = 2 z + 2 \cdot - 4$

Simplify the arithmetic:

$z = 2 z - 8$

Subtract from both sides:

$z - 2 z = (2 z - 8) - 2 z$

Simplify the arithmetic:

$- z = (2 z - 8) - 2 z$

Group like terms:

$- z = (2 z - 2 z) - 8$

Simplify the arithmetic:

$- z = - 8$

Multiply both sides by :

$- z \cdot - 1 = - 8 \cdot - 1$

Remove the one(s):

$z = - 8 \cdot - 1$

Simplify the arithmetic:

$z = 8$

10 additional steps

$z = 2 \cdot (- (z - 4))$

Expand the parentheses:

$z = 2 \cdot (- z + 4)$

$z = 2 \cdot - z + 2 \cdot 4$

Group like terms:

$z = (2 \cdot - 1) z + 2 \cdot 4$

Multiply the coefficients:

$z = - 2 z + 2 \cdot 4$

Simplify the arithmetic:

$z = - 2 z + 8$

Add to both sides:

$z + 2 z = (- 2 z + 8) + 2 z$

Simplify the arithmetic:

$3 z = (- 2 z + 8) + 2 z$

Group like terms:

$3 z = (- 2 z + 2 z) + 8$

Simplify the arithmetic:

$3 z = 8$

Divide both sides by :

$\frac{(3 z)}{3} = \frac{8}{3}$

Simplify the fraction:

$z = \frac{8}{3}$

3. List the solutions

$z = 8, \frac{8}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | z |$
$y = 2 | z - 4 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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

Latest Related Drills Solved