Inequalities (Pos/Neg): Is |a| + |b| > |a + b| ?

Based on the information provided,
|a| + |b| can be greater than |a+b| if either of a or b is a negative value. If both are either positive or negative, the two sides of the equation will be same and hence may still provide the answer to this DS question, if we cna ascertain the signs of a and b

Now,
1) a^2 > b^2 can be true for many scenarios, such as,
i) a and b are both positive and a > b
ii) Either a or b is negative and absolute value of a is greater than that of b
iii) Both a and b are negative and absolute value of a is greater than that of b

So the information provided is insufficient to answer the question

2) |a| * b < 0. One thing is tells is that b is -ve. The condition can be true for the following scnearios,
i) if a is -ve
ii) if a is +ve

But as mentioned above the question can be answered only if we know for sure that either one of the values is -ve (and o ther positive) or both are +ve or -ve.

In both statements, we get multiple possible values and hence cannot answer the question, so "E"

From statement (1), a^2 > b^2, we can gather that abs(a) > abs(b). However, this is not enough information to tell us if both are positive, both are negative, or one is positive and one is negative. In the case where both a and b share the same sign, abs(a) + abs (b) will equal abs(a + b). Take a minute to convince yourself of this. If they have opposite signs, though, then abs(a) + abs (b) will be bigger than abs (a + b). Again, take a minute to convince yourself of this. Basically, some canceling out happens in the a + b part if the signs are different, and you are left with a number that is smaller than the absolute value of either one.

Since we don't know the signs of a and b, statement 1 is not sufficient.

Moving on to statement (2), abs(a) x b > 0 tells us that b <0> abs(b) and b < 0. This STILL doesn't tell us the sign of a. We could have the situation where a < b< 0, or the situation where b < 0 <a> abs(b).

Assigning real numbers, what I am staying is we could have a = -4 and b = -3, or we could have b = -3 and a = 4. In the first case, abs(a) + abs (b) will equal abs(a + b). In the second case, abs(a) + abs (b) will be greater than abs(a + b).

Thus, the statements combined do not provide sufficient information and the answer is E.

Is |a| + |b| > |a + b| ?

(1) a^2 > b^2

(2) |a| × b < 0

Stem1:
Not suff as we can't say which one is -ve or +ve.

Stem2:
ab<0, meaning that b is negative. But a may be +ve or -ve. NOT SUFF.

1+2:
Still we are not sure whether a is +ve or -ve. NOT SUFF.

Answer is E.