l x l = x l x l = -x explain?

Question

lxl=x if x is positive
lxl=-x if x is negative

can someone explain this?

sorry the title is confusing

Answer 1

(hint: use the "pipe" character | instead of the letter l, that will be less confusing - it's usually above the \ key on a standard keyboard, but is sometimes in other positions.)

Sometimes it's useful to have an idea of the size of a number, without caring whether it's positive or negative. For instance, you may be interested in knowing how accurate some prediction is, withouth worrying about whether it was above or below the actual result. The absolute value function |x| is designed for this purpose.

|x| is defined, as you said, as x if x is positive (or zero) and as -x if x is negative. This has the effect of stripping off any negative sign that is present, so |3| = 3 and |-3| = 3. It should be easy enough to see that regardless of whether x is positive or negative, |x| = |-x|. You should also be able to see that |x| could also be defined as sqrt(x^2).

I'm not sure if your title represents an actual problem you're having trouble with, but if so, here's how to do it. Note that a = b = c means that a = b and b = c (and therefore also a = c). Verifying any two of these equations implies the third.

Suppose |x| = x |x| = -x. The easiest equality to start with is |x| = -x. If this is true, then x must be negative or zero, and x |x| = x (-x) = -x^2. This then gives -x^2 = -x which means that x = 0 or x = 1. But x is negative or zero, so the only solution is x = 0.

Answer 2

This is how the absolute value function is defined. Let's say x=-8, then it is negative so we use the rule |x|=-x=-(-8)=8. Now let's say x=8, then it is positive so |x|=x=8. Basically the absolute value function makes a number always postive regardless of how it began.