GMAT prep exam one question. I was unable to find this in the forms. I searched using first few words of the question as a search.

Which of the following inequalities has a solution set that, when graphed on the number line, is a single line segment of finite length.

a) X^4 ≥ 1
b) X^3 ≤ 27
c) X^2 ≥ 16
d) 2 ≤ |x| ≤ 5
e) 2 ≤ 3x + 4 ≤ 6

OA: E

Let's think about what is means to have a solution set that when graphed on the number line is a single line segment of finite length. That means that I must have a specific set of solutions that begins in one spot and ends in another, encompasses everything in the middle, and does not create two lines.

I can rule some of these out pretty quickly. With A, I have possible values equal to or above 1 and equal to or below -1. This fails on two counts--two separate lines plus they go on to infinity. Same thing with C--4 and greater numbers work as do -4 and lesser numbers.

B is very similar--anything 3 or below will work, on through negative infinity.

D looks good at first, but we have two solutions because of the absolute value signs; either 2 to 5 or -2 to -5 will work. Two lines so we have to toss that out.

E is the only one left. You can solve by first obtaining the x for 2 ≤ 3x + 4 then obtaining the x for 3x + 4 ≤ 6; in other words, split the equation into two pieces.

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Jamie Nelson
ManhattanGMAT Instructor

Thanks Jamie for your swift reply. I think my issue really here was that I did not understand what “finite length” really meant.

Oh, I see. Finite length means there is a particular starting and stopping point; you cannot have a line that goes on to infinity.

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Jamie Nelson
ManhattanGMAT Instructor

Hello Jamie,

I have a doubt related to this question:

when you solve the inequality in (e) you get -2/3 <=x <= 2/3
Here: <= (greater than equal to) (sorry about the format issue)


This should practically mean that x can lie within a certain range and y can have any value. Wouldn't that specify a region enclosed between two points instead of a line?

Hi erpriyankabishnoi,

This question is only talking about a number line. There is no Y component present or Y axis to think of. So just think of it in terms of plotting the solution on a straight horizontal line with 0 as the origin, all positive x's on the right, negative x's on the left.

Regards

Sunil

Thanks a lot, Sunil. This helps!

to the poster above, note that the question says “when graphed on the number line”; these words specify the one-dimensional interpretation explained by the previous poster.