I am studding modulus function at college. I dont understand this part can you explain each of these please?

І x a І =x - a for x ≥ a and I x- a I = -( x a ) = a-x for x<a

I x b I ≤ a ↔ - a ≤ x b ≤ a

І x b I ≤ a ↔ - a + b ≤ x ≤ a +b

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- November 7th 2009, 01:54 PMmike789help with modulus function
I am studding modulus function at college. I dont understand this part can you explain each of these please?

І x a І =x - a for x ≥ a and I x- a I = -( x a ) = a-x for x<a

I x b I ≤ a ↔ - a ≤ x b ≤ a

І x b I ≤ a ↔ - a + b ≤ x ≤ a +b - November 7th 2009, 02:25 PMProve It
These are basic definitions of the modulus function.

1. Remember that, by definition:

.

Now replace with .

What happens to the resulting inequalities?

2. It helps if you think of the modulus function as the "size" of whatever is inside it.

So is the same as saying

"the size of is never any more than ."

It then follows that

is equivalent to .

3. We saw from Q2 that

is the same as

.

Now add to everything in the inequality. What happens?