ABSOLUTE VALUE

Absolute Value in Algebra

Absolute Value means ...

... only how far a number is from zero:
"6" is 6 away from zero,
and "-6" is also 6 away from zero.
So the absolute value of 6 is 6,
and the absolute value of -6 is also 6

Absolute Value Symbol

To show you want the absolute value of something, you put "|" marks either side (called "bars"), like these examples:
|-5| = 5|7| = 7


So, when a number is positive or zero we leave it alone, when it is negative we change it to positive.
More Formal

This can all be written like this:
Absolute Value
This says: the absolute value of x equals:
  • x when x is greater than zero
  • 0 when x equals 0
  • -x when x is less than zero (this "flips" the number back to positive)
Here is an example:

Example: what is |-17| ?

Well, it is less than zero, so we need to calculate "-x":
- ( -17 ) = 17
(Because two minsus makes plus)

Useful Properties

Here are some properties of absolute values that can be useful:
|a| ≥ 0 always!
That makes sense ... |a| can never be less than zero.
|a| = √(a2)
Squaring a makes it positive or zero (for a as a Real Number). Then taking the square root will "undo" the squaring, but leave it positive or zero.
|a × b| = |a| × |b|
Means these are the same:
  • the absolute value of (a times b), and
  • (the absolute value of a) times (the absolute value of b).
Which can also be useful when solving
|u| = a is the same as u = ±a and vice versa
Which is often the key to solving most absolute value questions.

Example: solve |x+2|=5

Using "|u| = a is the same as u = ±a":
this: |x+2|=5
is the same as this: x+2 = ±5
Which will have two solutions:
x+2 = -5x+2 = +5
x = -7x = 3

Graphically

Let us graph that example:
|x+2| = 5
It is easier to graph if you have an "=0" equation, so subtract 5 from both sides:
|x+2| - 5 = 0
And here is the plot of |x+2|-5, but just for fun let's make the graph by shifting it around:
|x+2| - 5 = 0
Start with |x|then shift it left to make it|x+2|then shift it down to make it|x+2|-5
And you can see the two solutions: -7 or +3.

Absolute Value Inequalities

Mixing Absolute Values and Inequalites needs a little care!
There are 4 inequalities:
< >
less thanless than
or equal to
 greater thangreater than
or equal to

Less Than, Less Than or Equal To

With "<" and "" you get one interval centered on zero:

Example: Solve |x| < 3

This means the distance from x to zero must be less than 3:
-3 to 3
Everything in between (but not including) -3 and 3
It can be rewritten as:
-3 < x < 3
And as an interval it can be written as: (-3, 3)
The same thing works for "Less Than or Equal To":

Example: Solve |x| ≤ 3

Everything in between and including -3 and 3
It can be rewritten as:
-3 ≤ x ≤ 3
And as an interval it can be written as: [-3, 3]
How about a bigger example?

Example: Solve |3x-6| ≤ 12

Rewrite it as:
-12 ≤ 3x-6 ≤ 12
Add 6:
-6 ≤ 3x ≤ 18
Lastly, multiply by (1/3). Because you are multiplying by a positive number, the inequalities will not change:
-2 ≤ x ≤ 6
Done!
And as an interval it can be written as: [-2, 6]

Greater Than, Greater Than or Equal To

This is different ... you get two separate intervals:

Example: Solve |x| > 3

It looks like this:
|x| > 3
Up to -3 or from 3 onwards
It can be rewritten as
x < -3   or   x > 3
As an interval it can be written as: (-∞, -3) U (3, +∞)
Careful! Do not write it as
-3 > x > 3     no!
"x" cannot be less than -3 and greater than 3 at the same time
It is really:
x < -3   or   x > 3     yes
"x" is less than -3 or greater than 3

The same thing works for "Greater Than or Equal To":

Example: Solve |x| ≥ 3

Can be rewritten as
x ≤ -3   or   x ≥ 3
As an interval it can be written as: (-∞, -3] U [3, +∞)

S-BATCH