Alternatively, you can get rid of the mod sign and then proceed:

If |a+5|=|b+5|,

EITHER a + 5 = b + 5 (In this case a = b)

OR a + 5 = - (b + 5) (In this case a + b = -10)

1) b > 5 and a < 5
This statement tells you that a is not equal to b. Then, a + b = -10. Sufficient

2) b = 10
Here a could be 10 so a + b = 20 OR a + b = -10.
Not sufficient.

Answer (A)


Note that |a+5|=|b+5| gives you 4 cases but they are equivalent to 2 cases discussed above:
a + 5 = b + 5
a + 5 = - (b + 5)
-(a + 5) = b + 5
-(a + 5) = -(b + 5)
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Given that |a+5| = |b+5|. We can square on both sides, as both sides are positive and we get \(\to (a+5)^2-(b+5)^2 = 0 \to (a+b+10)(a-b)=0\)

From F.S 1, we know that \(a \neq{b}\). Thus, only \(a+b+10=0\) and a+b = -10. Sufficient.

From F.S 2, we get (a+20)(a-10) = 0. Thus, 2 values of a will give 2 different values of (a+b). Insufficient.

A.
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Hi All,

When DS questions look complex, they often hide subtle patterns. While you might not immediately 'see' what the pattern is, you can do a bit of work (TEST VALUES) and PROVE the pattern exists.

Here, we're told that |A+5| = |B+5|. We're asked for the value of A+B.

Fact 1: B>5 and A<5

This means that A and B CANNOT be the same number. Let's TEST VALUES and see what happens....

IF...
B = 6
|A+5| = 11

Since we were told that A<5, A can only have one solution....A = -16 (since |-16+5| = |-11| = 11).
The answer to the question is 6 + (-16) = -10

IF...
B = 7
|A+5| = 12

Since we were told that A<5, A can only have one solution....A = -17
The answer to the question is 7 + (-17) = -10

We were NOT told that A and B had to be integers though, so I'll try one more TEST....

IF...
B = 5.5
|A+5| = 10.5

Since we were told that A<5, A can only have one solution....A = -15.5
The answer to the question is 5.5 + (-15.5) = -10

The answer is ALWAYS -10
Fact 1 is SUFFICIENT

Fact 2: B = 10

This tells us nothing about the value of A.
Fact 2 is INSUFFICIENT.

Final Answer:

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The question can also be solved by basic interpretation of the concept of modulus.

Interpreting the Given Info
We know that modulus denotes the magnitude of the distance of a number from another number. So for |a + 5| = | b + 5|, we know that distance of a and b from -5 is the same. Thus there are two possible cases:

I. a = b. In this case a and b are at the same distance from -5 and thus a + b can take any value




II. a and b are on the opposite sides of -5 and equidistant from -5. Let's assume the distance be x. So a = -5 -x and b = -5 +x or vice versa. in both the cases we observe that a + b = -10.




In other words the question asks us to find if a = b?.

Statement-I
St-I tells us that b > 5 and a < 5 i.e. both are distinct numbers. So we know that a + b = -10.

Hence st-I is sufficient to answer the question

Statement-II
St-II tells us that b = 10. However it does not tell us anything about a. Here a = 10 or a = -20.
Hence st-II is not sufficient to answer the question.

Hope this helps

Regards
Harsh
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Quite Simple Ques by my approach

|a+5| = | b+5|
1) b>5 and a<5

let us say b=6
=> |a+5| = 11
a+5 = +11 , a=6 ( not possible as a<5)
a+5 = -11 , a=-16

a+b = -16 +6 =-10
if b=7 then a= -17 => a+b =-10
A/D can be answer

2) b=10
|a+5| =15
a=10 or a=-20
rejected

A is the answer
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Veritas Prep Official Solution:


This question is a great candidate for number picking. If you call b 10, then the second absolute value is 15, meaning that a must be -20 for the first absolute value to also be 15. Thus, a + b could be -10. Then try b = 15. In order to keep the second inequality equal, you’ll have to drop a by 5 to -25, again making the value of a + b = -10. This will hold for all potential values consistent with statement 1, so statement 1 is sufficient.

Additionally you can solve this algebraically with the information in statement 1. If b is positive then you know that |b+5|=b+5
. Logically the only way a + 5 could equal b + 5 if a < 0 is if the entire expression on the left is negative. Therefore you know that - (a + 5) = b + 5 and -a -5 = b + 5 and a + b = -10. Sufficient.

Statement 2, however, is not sufficient. Knowing that b = 10 means that either a + 5 = 15 or a + 5 = -15, and so with two possible values of a but one exact value of b you cannot solve for a + b, and the correct answer is A.
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