If |a + 5| = |b + 5| what is the value of a + b?

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.

Although you mentioned 4 scenarios , why did just consider 2 of them prior to moving to answer choices

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:

GMAT assassins aren't born, they're made,
Rich

The first and the fourth cases give us the same equation:

I. a + 5 = b + 5

IV. -(a + 5) = -(b + 5)
Multiply the equation by -1 on both sides to get
(a + 5) = (b + 5)

The second and third cases give us the same equation:

II. (a + 5) = -(b + 5)

III. -(a + 5) = (b + 5)
Multiply the equation by -1 on both sides to get
(a + 5) = -(b + 5)

So actually you have only 2 distinct equations.

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

nice explanation, thank you

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

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.

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.

If we square both the sides, the it will also give us a+b = -10. Where am I wrong?

I think this question doesn't require Data statements to solve the problem.

Consider |a+5|=|b+5|
Squaring both the sides,
\((a+5)^2=(b+5)^2\)
Or, \(a^2+10a+25=b^2+10b+25\)
Or, \(a^2-b^2=-10 (a-b)\)
or, \(a+b=-10\) which is what the question asks for.

This problem requires "a+b" value and isn't concerned with values of a or b. Hence determining them using (1) b > 5 and a < 5 ;(2) b = 10 seems meaningless

PS:The concept of Data Sufficiency(DS) in GMAT is the find out whether statements(supporting data) are sufficient to determine the answer. But here I statement became unnecessary or superfluous data in determining the answer. I believe GMAT doesn't ask the above question under the DS category.

Hi santosh93,

The initial information in a DS prompt is NEVER enough to conclusively answer the question - so if you think that you are facing a prompt in which you don't need any additional information, then it's likely that you are making some type of logic mistake while thinking about the prompt.

By working through the math steps that you've listed, you've fundamentally changed the original equation (and 'divided out' a potential solution). Consider the following two examples, based on the original equation:

|A + 5| = |B + 5|

IF.... A = 1 and B = 1, then |6| = |6| and the answer to the question is 1+1 = 2

IF.... A = 5 and B = -15, then |10| = |-10| and the answer to the question is 5 + (-15) = -10

Based on just the original equation, there is clearly more than one potential answer (so the assumption that the answer must be -10 is incorrect - and you need more information to determine whether there is just one answer or multiple answers).

GMAT assassins aren't born, they're made,
Rich

Hi EMPOWERgmatRichC
Thank you for the reply. Now I understand where I possibly went wrong while solving the original equation. I believe it was this step
\(a^2-b^2=-10 (a-b)\)
instead of dividing out a potential solution, I should consider it as a possible answer.
Solving the above I will end up having
\(a=b\) or Or, \(a+b=10\)

Hence I would require support statements to concretely answer the problem.

Once again thank you very much.

Can I say that |a + 5| = |b + 5| is equal to |a| + |5| = |b| + |5| ?

No you cannot. For example, if a=1, b=-11, |a+5|=|b+5|=6 but |a|+|5|=6 and |b|+|5|=16

In general,

1. Always true: |x + y| ≤ |x|+ |y| , note that "=" sign holds for xy >0 (or simply when x and y have the same sign);

2. Always true: |x – y| ≥ |x|-|y| , note that "=" sign holds for xy > 0 (so when x and y have the same sign) and |x| > |y| (simultaneously).

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