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