Absolute value - MODS

Question

Please help in resolving the confusion between the two problems below:-

Q1. If -|x+1| = b, where b is a non-zero integer, which of the following statements must be true?
I. b < 0

After solving the below steps i am not sure what to do

Given, -|x+1| = b
|x+1| = -b

After opening the mods, as per concept, we get two values one positive and another negative:-
(1) x+1 = -b and (2) x+1 = b

But as per explanation, the left hand side (modulus) is never negative, we can infer that “-b” on the RHS is positive.

the left hand side (modulus) is never negative [this one seems to be a bit confusing]

Another question is from MGMAT :-
22 - |y+14| = 20
|y+14| = 2
Now as per explanation given in the book there are two cases one in which expression is positive and one in which it is negative.
(1) y+14 = 2 and (2) y+14 = -2

What is the difference between in above 2 questions ? This small doubt is creating a lot of problem while solving questions.
Will really appreciate if anyone can help.

Thanks & Regards
Sachin

MartyMurray · Answer

sachin_yadav wrote:Please help in resolving the confusion between the two problems below:-

Q1. If -|x+1| = b, where b is a non-zero integer, which of the following statements must be true?
I. b < 0

After solving the below steps i am not sure what to do

Given, -|x+1| = b
|x+1| = -b

After opening the mods, as per concept, we get two values one positive and another negative:-
(1) x+1 = -b and (2) x+1 = b

But as per explanation, the left hand side (modulus) is never negative, we can infer that “-b” on the RHS is positive.

the left hand side (modulus) is never negative [this one seems to be a bit confusing]

An absolute value of anything is always positive or 0. It is the distance from 0. So if x is not zero, |x| is always positive. If z+2 is not 0, then |z+2| is always positive. |-1| is positive 1. |0| = 0.

So |x+1| is always positive or 0. So if -b = |x+1|, and b is non zero, then -b must be positive.

On the other hand, what is within the || can be negative. x+1 could be positive or negative.

So if -b is positive, then it may be that x+1 is positive and equal to -b or, if x+1 is negative then -(x+1) = -b, alternatively, x+1 = b.

It looks as if you basically get this. Maybe you just thought they were saying that x+1 is always positive, when actually they were saying that |x+1| is always positive.

So if |x+1| = -b, and b is not 0, then -b has to be positive and b < 0.

sachin_yadav wrote:Another question is from MGMAT :-
22 - |y+14| = 20
|y+14| = 2
Now as per explanation given in the book there are two cases one in which expression is positive and one in which it is negative.
(1) y+14 = 2 and (2) y+14 = -2

What is the difference between in above 2 questions ? This small doubt is creating a lot of problem while solving questions.
Will really appreciate if anyone can help.

Thanks & Regards
Sachin

Ok we know that |y+14| = 2.

The thing is that if y+14 is non zero, then putting y+14 into || creates a positive number, whether or not y+14 itself is positive.

y+14 could be equal to 2 or y+14 could be equal to -2, and as soon as we put y+14 into ||, the result will always be positive. So either way |y+14| = 2.

So the converse is that when we know that |y+14| = 2, all we can tell is that either y+14 = 2 or y+14 = -2.

So really there is no difference between what is going on in the first question and what is going on in the second.

Rich.C@EMPOWERgmat.com · Answer

Hi sachin_yadav,

Sometimes when dealing with a Quant concept that seems a bit weird or abstract, it helps to TEST VALUES so that you can see how the concept "works"

In your first question, we're told...

- |X+1| = B

Let's deal with this on it's own first....

Absolute value signs turn "negative results" into "positive results", so |X+1| will be greater than or equal to 0 (it just depends on what you plug in for X).

If...
X = -1, then |-1 + 1| = 0

X = -2, then |-2 + 1| = |-1| = 1

X = 4, then |4 + 1| = |5| = 5

Next, we're told that B is a NON-ZERO INTEGER. This adds a restriction to our prior work. Since B CANNOT be 0, then X CANNOT be -1.

Finally, we have a "minus sign" in front of the absolute value symbol. In this question, since we know that B cannot = 0 and we know that |X+1| is greater than or equal to 0 AND now we have a minus sign in front of |X+1|....

The left side of the equation MUST be negative, which means that B MUST be negative (B < 0).
----------------

In your second question, you correctly simplified the equation down to the following:

|Y+14| = 2

Since absolute values turn "negative results" into "positive results", there are probably 2 values for Y that will fit this calculation: one that yields a NEGATIVE result and one that yields a POSITIVE result.

If...
Y = -12, then |-12 + 14| = |2| = 2

Y = -16, then |-16 + 14| = |-2| = 2

The big take-away from all of this is that when you're dealing with an Absolute Value sign, there is probably going to be more than one answer that fits the given equation (or inequality), so be ready to do a bit more work to find the other answers.

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

Contact Rich at [email protected]

GMATGuruNY · Answer

I would approach the two problems differently.

When the question stems asks "Which of the following must be true?," one strategy is to try prove that each statement does NOT have to be true.

sachin_yadav wrote:If -|x+1| = b, where b is a non-zero integer, which of the following statements must be true?
I. b < 0

Strategy: Try to prove that it does NOT have to be true that b<0.
In other words, test whether it's possible that b>0.

Case 1: b=1
-|x+1| = 1
|x+1| = -1.
Since absolute value cannot be negative, it is not possible that b=1.

Case 2: b=10
-|x+1| = 10
|x+1| = -10.
Since absolute value cannot be negative, it is not possible that b=10.

Case 3: b=1/2
-|x+1| = 1/2
|x+1| = -1/2.
Since absolute value cannot be negative, it is not possible that b=1/2.

By now we should be convinced.
It is not possible that b>0.
Thus, since b is nonzero, it must be true that b<0.

Another question is from MGMAT :-
22 - |y+14| = 20
|y+14| = 2

Whereas the first problem offers an inequality yielding an infinite number of options for x and b, this problem -- an equation with only one variable -- can actually be solved.

Case 1: No signs changed
y+14 = 2
y = -12.

Case 2: Signs changed on ONE SIDE
y+14 = -2
y= -16.

Thus, there are two solutions:
y=-12 and y=-16.

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