CM12 wrote:
Quote:
These three posts explain these points comprehensively. Please check:
Hi
KarishmaB - I've been going through this thread/your articles (all great resources/posts, so thank you!). One bit of confusion as I'm still wrapping my head around when Division of the roots doesn't change anything, as well as the situations where the inequality includes an "=" sign. When I read the 3rd post you link, I cannot tell what the inequalities are (all I see are "?" where the signs should be), and can't grasp this topic.
As well, I don't see much detail on why the division aspect has no impact on the values. Would you be able to elaborate just a little, or link another resource if easier?
Thanks so much, everything else makes perfect sense so far!
CM12Hope you understand that the solution of say (x-a)(x-b)(x-c) < 0
is all about the sign of the factors (x-a), (x-b) and (x-c).
(x-a)(x-b)(x-c) < 0 only means that (x-a)(x-b)(x-c) must be negative. Which means that either exactly one of these factors or all three must be negative and our entire method is based on that one concept.
Now notice that if instead the factors are written as (x-a)(x-b) / (x-c) does it change anything regarding the sign of the expression based on the signs of the factors?
Consider two numbers a and b.
The sign of both ab and a/b will depend on the sign of a and sign on b.
If a and b are both positive, ab and a/b both are positive too.
If a is positive and b is negative, ab and a/b both are negative.
If a is negative and b is positive, ab and a/b both are negative.
If a and b are both negative, ab and a/b both are positive.
It doesn't matter whether a and b are multiplied or divided, the sign of ab and a/b will always be the same.
The only difference between ab and a/b is that in a/b, b cannot be 0 because division by 0 is not allowed.
So solution of \(ab \leq 0\) includes a = 0 or b = 0 (because the expression can be 0 too now)
but solution of \(\frac{a}{b} \leq 0\) includes only a = 0 (b cannot be 0)
So solution of (x-a)*(x-b) < 0 is the same as the solution of (x-a) / (x-b) < 0.
But solution of \((x-a)*(x-b) \leq 0\) includes x = a, x = b but solution of \(\frac{(x-a)}{(x-b)} \leq 0\) includes only x = a.
This is the concept in brief.
I have discussed it in more detail in my inequalities and absolute values module here:
https://anglesandarguments.com/study-moduleThat helps a lot and makes sense, thank you again. I was definitely over thinking the division aspect.
. Based on this and in regard to the 4th lesson covered early in this thread around roots & inequalities, how come \(\sqrt{x} < 10\) results in 0 < x < 100 when squared? Mainly, why must x be greater than 0, and not > or = 0?