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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- Nov 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 - Nov 7th 2009, 02:25 PMProve It
These are basic definitions of the modulus function.

1. Remember that, by definition:

$\displaystyle |x| = x\textrm{ if }x > 0, 0 \textrm{ if }x = 0, -x \textrm{ if }x < 0$.

Now replace $\displaystyle x$ with $\displaystyle x - a$.

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 $\displaystyle |x - b| \leq a$ is the same as saying

"the size of $\displaystyle x - b$ is never any more than $\displaystyle a$."

It then follows that

$\displaystyle |x - b| \leq a $ is equivalent to $\displaystyle -a \leq x - b \leq a$.

3. We saw from Q2 that

$\displaystyle |x - b| \leq a$ is the same as

$\displaystyle -a \leq x - b \leq a$.

Now add $\displaystyle b$ to everything in the inequality. What happens?