Math Help - help with modulus function

1. help with modulus function

I am studding modulus function at college. I don’t 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

2. Originally Posted by mike789
I am studding modulus function at college. I don’t 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
These are basic definitions of the modulus function.

1. Remember that, by definition:

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

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

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

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

It then follows that

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

3. We saw from Q2 that

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

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

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