Is r = s ? (1) -s <= r <= s (2) |r| >= s

For people who look at mod from the number line perspective, you can also do this:

(1) -s<=r<=s
On the number line, r lies between s and -s. i.e. if s = 4, -4 <= r <= 4.

(2) |r|>=s

This says that r >= s or r <= -s i.e. r is to the right of s or left of -s (or the same as s or -s). If s = 4, r >= 4 or r <= -4

Together, we can say that r must be equal to either s or -s i.e. if s = 4, r = 4 or -4. We still cannot say whether r = s.
Hence not sufficient.

Answer (E)

(1) -s<=r<=s not suffice
(2) |r|>=s
i.e. r >= s for positive s, or r < = -s for negative s , not sufficient

from both we get, r = s and r = -s
therefore, We cannot determine

Many thanks karishma

Discussed here: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653870

Dear Karishma:
How is he translating from:
(1) -s<=r<=s to
to:
/r/>=s
Do you have any helpful links for this topic? Im struggling heavily with it
Thank you

Hello reto

What is module:
Module \(|r| = r\) than \(r > 0\)
and \(|r| = -r\) than \(r < 0\)
so \(|2| = 2\) and \(|-2| = 2\)
(just for simplifying you can think about module as about dropping minus from number, but for harder tasks you should use operation described above)

And about our task:
We can make example from this inequality \(−s≤r≤s\) --> \(-4 ≤ 2 ≤ 4\) or \(-4 ≤ -2 ≤ 4\)
So this inequality will be true for \(r\) and for \(r *-1\)

So we can change \(r\) in this inequality on \(|r|\) and this inequality still be true --> \(-4 ≤ |2| ≤ 4\)
And from this part we can drop left side of inequality and write it as \(|2| ≤ 4\)

That's how we transform \(-s≤r≤s\) to \(|r|≤s\)

Theory about modules: math-absolute-value-modulus-86462.html

Hey Harley!
Good to see you here! Thanks for your explanation - i really gotta study hard for that quant stuff!
Thanks and all the best

I am guessing you have already got the logic of how -s<=r<=s translates to |r| <= s.

Keep these basic relations in mind:

|r| <= s
implies -s <= r <= s

|r| >= s
implies either r <= -s or r >= s

r, the term inside the absolute value could be anything.

|x - 4| <= 4
implies -4 <= x - 4 <= 4
Add 4 to the inequality to get:
0<= x <= 8
etc

Given : Nothing
DS: r = s

Statement 1 : \(-s \leq r \leq s\)
Not sufficient
\(|r| \leq s\)

Statement 2: \(|r|\geq s\)
Not sufficient

Combined : |r| = s
r = +/- s
Not sufficient

Answer E

(1) \(-s \leq r \leq s\) gives several values for r. So Insufficient.

(2) \(According \ to \ absolute \ value \ rule\) \(|r|\geq s\) is same as \(-s \leq r \leq s\). So Insufficient.

Considering both options gives us the same result. So C insufficient too.

The answer is \(E\)

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