# 2 - Absolute value

|3| = 3
|4| = 4
|50| = 50
|-3| = 3
|57| = 57
|-1000| = 1000
|0| = 0
|-789| = 789

The last identity reads : "the absolute value of -789 is 789".
The absolute value of a number is always a positive number.
We can remove the bars the writing of |x|, according to the following rule : you have to distinguish two cases : if x is positive, then |x| = x, and if x is negative, then |x| = - x ( |-9| = - (-9) = 9). If there is a more complicated number between the absolute value bars, you do the same way, |2x - 4| = 2x - 4 if x > 2 (2x - 4 > 0) and |2x - 4| = -2x + 4 if x < 2 (2x - 4 < 0).

How to solve |x - 5| < 3.
You have to consider two cases, so you have to write down a system of equations :

Hence .

You can also solve this kind of equations thanks to a graph. You have to know that |x - y| < a means "the distance between x and y is smaller than a". In that case it is "the distance between x and 5 is smaller than 3". So you place the number 5 then you count 3 on both sides. As it is the sign "smaller than" and not "smaller than or equal to", 2 and 8 are not solutions of the system.

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Absolute values

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