Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

z = - \frac{25}{4}, \frac{25}{6}

z=-\frac{25}{4} , \frac{25}{6}

Mixed number form:

z = - 6 \frac{1}{4}, 4 \frac{1}{6}

z=-6\frac{1}{4} , 4\frac{1}{6}

Decimal form:

z = - 6.25, 4.167

z=-6.25 , 4.167

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
$| 5 z | = | z - 25 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 z \| = \| z - 25 \|$
$x = + y$	$(5 z) = (z - 25)$
$x = - y$	$(5 z) = - (z - 25)$
$+ x = y$	$(5 z) = (z - 25)$
$- x = y$	$- (5 z) = (z - 25)$

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:

$\| x \| = \| y \|$	$\| 5 z \| = \| z - 25 \|$
$x = + y$ , $+ x = y$	$(5 z) = (z - 25)$
$x = - y$ , $- x = y$	$(5 z) = - (z - 25)$

2. Solve the two equations for z

5 additional steps

$5 z = (z - 25)$

Subtract from both sides:

$(5 z) - z = (z - 25) - z$

Simplify the arithmetic:

$4 z = (z - 25) - z$

Group like terms:

$4 z = (z - z) - 25$

Simplify the arithmetic:

$4 z = - 25$

Divide both sides by :

$\frac{(4 z)}{4} = \frac{- 25}{4}$

Simplify the fraction:

$z = \frac{- 25}{4}$

6 additional steps

$5 z = - (z - 25)$

Expand the parentheses:

$5 z = - z + 25$

Add to both sides:

$(5 z) + z = (- z + 25) + z$

Simplify the arithmetic:

$6 z = (- z + 25) + z$

Group like terms:

$6 z = (- z + z) + 25$

Simplify the arithmetic:

$6 z = 25$

Divide both sides by :

$\frac{(6 z)}{6} = \frac{25}{6}$

Simplify the fraction:

$z = \frac{25}{6}$

3. List the solutions

$z = - \frac{25}{4}, \frac{25}{6}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 z |$
$y = | z - 25 |$
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