If b < 1 and 2x - b = 0, which of the following must be true?

Hello All,

X < 1/2 is correct, two answer choices seems to be correct
B.x<-2
D.X<3
It is not given than X is a postive or negative, integer or fraction.
In my opinion, if we consider D it can have possible answer as X = 2 or X = 1, but X has to be less than 1/2.
If we consider X < -2 for all values of X, X < 1/2 holds true.
Hence my answer was B.

Please feel free to correct me.

REgards,
Pritish

Hi Pritish,

Question says : if b< 1 and 2x-b = 0 , which of the following must be true?or in simple words ,
"if x<1/2 , which of the following must be true?"

Option B doesnt hold good for any value of x where,
\(-2 <=x <1/2\)

Consider for example, if x =0,
x<1/2 is true but x <-2 is not true
Hence B can not be the answer.

On the other hand, for option D as others have pointed out correctly. Since x <1/2 and 1/2 <3
this implies that x <3 . This would be true for any value of x that satisfies x<1/2.

hope it helps.

As you mentioned
"Consider for example, if x =0,
x<1/2 is true but x <-2 is not true
Hence B can not be the answer."

If we chose option B X can never be equal to 0, but if X<3 there is a possibility that X can be 0. Right?

Hi Pritish,
You are getting confused here -
the reasoning follows through question stem first -
question says if x<1/2 then which of the following must be true -
This means x<1/2 is taken for granted. that is our scope. period.

now within this scope we need to find the answer. You dont choose option first and then try to fit in question, but u read option first, define the limit and then consider and choose options.
Hope it is clear. Now click kudos

Notice that we are asked "which of the following MUST be true?" not COULD be true.

Now, we know that x<1/2, thus x<3 is always true.

Is x<-2 always true? No, if x=0, then x<-2, won't be true, therefore this option is not always true.

For more on Must/Could be true questions check: search.php?search_id=tag&tag_id=193 (more than 100 questions).

Hope it helps.

So I was getting confused here especially in comparison to this problem

If a^5 ≤ a, which of the following must be true?

I. –1 ≤ a ≤ 0
II. a=0
III. 0 ≤ a ≤ 1

A. None of the above
B. I only
C. II only
D. III only
E. I and III only

Here the answer is A, because of the two possible options II and III, III cannot be right because we can think of -2 and find it to satisfy the original condition and -2 doesn't fall in this range. Although II is true, there are other numbers that satisfy the condition as well.

When I looked at this problem, I am like the correct answer should be x <= 1/2. But those choices were there. The answer was totally unexpected X<3, cause I though how about x = 1, 2 or some other value between 1/2 and 3. These values don't satisfy the condition, then how is it true.
This is what I have come up with, especially dealing with inequality problems, try to find a number (outside the range given in the answer choice) and see if it satisfies the question stem. If there is none, then you have the right answer. So for eg. looking at x<3, there is no value of x>=3 that would satisfy the Question stem and hence incorrect.

This is quite tricky to wrap your head around. Very tricky! hopefully this will help me in the future.

Hey nphatak...Greetings!!!


I'm a little confused about the example you gave and specially the answer choices. Is this question a standard GMAT question. I don't think the GMAT tries to trick you this way or is it possible

it's not a tricky question , trust me .
whenever you face such inequalities , first thing to do is to draw their roots on number line .

\(a^5-a <=0\)
\(a(a^4-1) <=0\)

root are -1, 0 , 1 now as shown in attached image you can find the regions where this inequality holds .
and clearly say that answer is A .

Well you see its not finding the roots that is the problem. I can find the conditions under which the inequality will hold. Its the answer choices that I said were confusing. The answer choice a=0 is true, the inequality will be 0. And when you are asked what must be true, you are like..yeah at a = 0 this is true. But the question is not if its true at one value, the question is which answer choice covers all the possible values. Not one! I am probably not able to explain what the confusion is.

If a^5 ≤ a, which of the following must be true?

I. –1 ≤ a ≤ 0
II. a=0
III. 0 ≤ a ≤ 1

A. None of the above
B. I only
C. II only
D. III only
E. I and III only

So when you solve the inequality, you get a <= -1 OR 0 <= a <= 1.
a is either less than -1 or it is between 0 and 1.

Let's see what each statement says.

I. –1 ≤ a ≤ 0
This says that a must be between -1 and 0. True or False? False

II. a=0
This says that a must be 0. True or False? False. a is either less than -1 or it is between 0 and 1.

III. 0 ≤ a ≤ 1
This says that a must lie between 0 and 1. True or False? False. a is either less than -1 OR it is between 0 and 1.

Now, think if there were another statement
IV. a < 2
This says a must be less than 2. True or False? True. a is either less than -1 OR it is between 0 and 1. In any case, it will always be less than 2.

By the way, it is not a trick. It is based on logic and can easily be tested on GMAT.

Thanks a lot Karishma!
So the correct answer choice should cover all the possible values of a, right? The way I was interpreting this question was, which of this must be true = any subset of all the possible values has to be true and I marked that answer. Its just a failure in my understanding these kind of questions.

Manipulating the equation we have 2x = b, thus:

2x < 1

x < 1/2

Thus, x must be less than 3.

Note that the correct answer x < 3 may be confusing. Here’s the logic: we determined algebraically that x must be less than 1/2. Thus, some possible values for x are 1/4 , 0, -⅔, -5, and so on. Note that each of these values is, indeed, less than 3 (which is answer choice D). In fact, any value of x that satisfies x < 1/2 will ALSO satisfy x < 3. Hence, answer choice D is correct.

Answer: D

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.