Answer is C.
This DS question expects an answer in Yes or No.
Consider the modulus expression |x - a|. As we know, Mod of any number gives out a positive value. But what is inside the Mod ("x - a" in this case) may have any sign or even have value zero.
Ex: "x - a" will be positive for x > a, negative for x < a and equal to 0 for x = a.
Two important things to note here. First, x = a is the point where sign of the expression gets reversed. Second, a negative value comes out of the Mod as positive, so when "x - a" is negative |x - a| = -(x - a), just as | -3 | = -(-3) = 3.
Now, coming to provided statements:
Statement 1:
Given b = 3.
So the given question becomes, Is |2a - 9| < |a - 3| + |a - 6| ?.
Now as mentioned above about the modulus expression, each of the three expression inside Mod will change sign based on value of a (in this case). The expression in question a has 3 sign reversal points: 9/2 , 3 and 6 respectively for "2a - 9", "a - 3" and "a - 6".
On number line:
<----------- 3 -------------- 9/2 -------------- 6 ----------------->
This is quite understandable that for each expression`s sign change, the complete equation will change.
For a < 3 (First case of Statement 1),
All the above "2a - 9", "a - 3" and "a - 6" will be negative: so, |2a - 9| = - (2a - 9),
|a - 3| = -(a - 3) and |a - 6| = -(a - 6). And the complete expression will be:
L.H.S = 9 - 2a
R.H.s = 3 - a + 6 - a = 9 - 2a.
Here we can see that for any a < 3, L.H.S = R.H.S. So according to First case of Statement 1, answer to asked question |2a - 9| < |a - 3| + |a - 6| ? is NO.
Again, for 3 < a < 9/2 (Second case of Statement 1),
Putting the values according to change signs (please try seeing which of the three have need to change sign),
L.H.S = 9 - 2a
R.H.S = 3.
Just put any value between 3 and 9/2 for a. L.H.S is always less than R.H.S. So according to Second case of Statement 1, answer to asked question |2a - 9| < |a - 3| + |a - 6| ? is YES.
We can stop here, since 2 different conditions for statement 1 gives contradicting results.
If this was not the case, we would have to check for all the ranges of values for a. For given situation statement 1 is clearly INSUFFICIENT.
Statement 2:
This also insufficient because we cannot decide for the signs of expressions inside Mod. INSUFFICIENT.
Statement 1 + 2:
b = 3 and a < b => a < 3.
From our previous analysis we can see that for a < 3, we have one undoubted answer (No) for question Is |2a - 9| < |a - 3| + |a - 6| ?
So, Statement 1 + 2 is sufficient to answer the question.
Hence C is correct.
**It looks lengthy process, but just because it is explained. With practice you can reduce time taken to solve below 2 mins.
Hope this helped.!!!!