Is b<50 (1) The square of (b+x) is less than 2500 (2) The square of (

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Done. Thank you.

at x= 0 we can have largest value for b, which is smaller than 50 and even if we consider negative value for b it will be smaller than 50 so why option A is not sufficient?

Simply because x can be -ve or +ve value....

Here's an approach testing specific cases:

(1) This one simplifies to -50 < b + x < 50.

Case 1: b = 0, x = 0, answer is "yes"
Case 2: b = 100, x = -100, answer is "no"

Not sufficient.

(2) Well, 1500 is a little less than 1600, so the square root should be a little less than 40. If I need to get it more precisely, I can do that later, but for now I'll say '~40' (approximately 40).

~-40 < b - x < ~40

Case 1: b = 0, x = 0, answer is "yes"
Case 2: b = 100, x = -100, answer is "no" (Try to notice chances to reuse cases! That can speed things up.)

Not sufficient.

(1 + 2)
Now I have two inequalities:

~-40 < b - x < ~40
-50 < b + x < 50

Notice that the x and -x will cancel if I add these inequalities together. You can always add any two inequalities whose inequality signs are pointing in the same direction. So, we get this:

~-90 < 2b < ~90
~-45 < b < ~45

We already decided that '~45' is less than 45, since 1500 is less than 1600. So, b is definitely less than 45, and the two statements combined are sufficient.

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