I can say with reasonable certainty that until and unless you are a quant jock, you hate inequalities (especially when they present themselves in DS questions). I can also say the same thing about absolute value (perhaps, not to the same extent!). So I can imagine what you feel when you come across a DS question with both absolute value and inequalities! I am also certain that a small part of you, a very small part indeed, is secretly thrilled to see such a question because it implies that you are doing well in the exam and the software is getting jittery and trying to give you harder and harder questions. But wouldn’t it be something if you could crack the question in a minute and then say, “What else you got?”

It is not very difficult to do that, in my opinion. GMAT is not about solving complicated equations/inequalities. It is about using your bare fundamentals, quietly, inconspicuously… I am reminded of Harry Potter and the Half-Blood Prince here (Yes, I am a Harry Potter fan and this time, I cannot blame my little daughter for it!). Dumbledore and Harry go to some cave to get a Horcrux and Dumbledore quietly works his way through, feeling around for magic. At that point, the author says, “He had never seen a wizard work things out like this, simply by looking and touching; but Harry had long since learned that bangs and smoke were more often the marks of ineptitude than expertise.” I think it is quite an appropriate analogy. GMAT is not about booms and bangs and fancy equations; it is more ethereal; you have to be more in touch with your concepts…

Anyway, I don’t want you to run away if you came here looking for some solid fundamentals. Let me start by saying that inequality just implies less than or greater than. Don’t try to read too much into it. Absolute value stands for the distance of a point from 0 on the number line. When I say |x| = 5, it means x is a point at a distance 5 from 0 on the number line. Take a minute to go through it again: x is a point at a distance 5 from 0 on the number line. Which points are at a distance 5 from 0 on the number line?

Points 5 and -5. So x can take two values: 5 or -5. Similarly, if |x| = 8, x can take two values x = 8 or x = -8.

Now, what if you have something like |x-3| = 5? What does this mean? The distance you are looking for is still 5. The only difference is that the distance is from the point 3 now.

Let’s look at the figure to see what values x can take.

In the last example above, we say that twice the distance from 1.5 is 5. So we want the point that is 2.5 away from 1.5.

If this makes sense to you, we could include an inequality sign here. Things will still be no different. For example, what values can x take if |x-3| < 5. We need those points where distance from point 3 is *less than* 5. At all the points depicted by the green line in the diagram below, the distance from 3 is less than 5. Point 4 is at a distance 1 away from 3, point 7 is at a distance 4 away from 3, 0 is at a distance 3 away from 3 and so on… Point 8 is at a distance 5 away from 3 so all points lying between 3 and 8 are at a distance less than 5 from 3. Point 3 is at a distance 0 from 3 so it is also included but 8 is not since the inequality doesn’t have an equal to sign.

This is an extremely efficient way of working with inequalities + absolute value.

*Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!*

Hey Karishma, this is indeed an interesting approach to inequalities and modulus

great post!

Really helpful article. Thanks Karishma

How does this work for |3-x| say where x is -ve?

Good article, Karishma. You are a Karishma:-)

I knew about this approach – But, badly needed to refresh. So, if anyone wants to refresh this approach, read this article from bottom to top.

You will try to plug in the gaps. An alternative approach to learning !!

|3-x| is the same as |x-3|.

Think of it this way:

|x| = |-x| for every value of x. e.g. |6| = |-6| = 6

|x-3| = |-(x-3)| = |3-x| for every value of x

So whenever you have |3-x|, just re-write this as |x-3| and continue as before.

Awesome post!!!

Perfect. To someone who’s completely floundering with inequalities, this comes as a Godsend…thanks for putting this down. I now get it. At least the basics…

With any luck, you’ll put up more substantial problems involving this topic…

Awesome post…………..thanks

thank you Karishma. That was a good way to absorb the concept. I will practice on this to make it solid.

Hi:

Quick question on the inequality abs (2x -3) >5 I don’t get how you get 2 * abs (x -1.5) . Please explain and also where the x = -1 is coming from. I think you are factoring…

Hi Sagnik,

In this ineeuality, |2x -3| >5:

As 2 is a +ve quantity, u can take it was common factor to isolate x.

That makes the eqn 2 | x-1.5|>5 .

Note, here we are taking 2 as common factor as u would do if it was (2x-3).

Now we can further simplify the inequality by dividing by 2 on both sides

|x-1.5| > 2.5

Hope it makes things clear.

Hi Sagnik,

In this inequality, |2x -3| >5:

As 2 is a +ve quantity, u can take it as common factor to isolate x.

That makes the eqn 2 | x-1.5|>5 .

Note, here we are taking 2 as common factor just as u would do if it was (2x-3).

Now we can further simplify the inequality by dividing by 2 on both sides

|x-1.5| > 2.5

Hope it makes things clear.

Hi Karishma,

Thanks a lot for this – i remember learning it exactly like this in school . I was struggling with inequalities while preparing for GMAT and now it all came rushing back ! :) Thanks a lot!

Excellent Karishma!

Thanks for taking up this topic :)

Karishma –

I absolutely love the way I re-learn topics, such as inequalities, from your blogs, allowing me to solve related problems in a far more efficient manner simply by visualizing them. Thank you so much. Material such as your blog posts will get me to a 750.

Do you have an index with all your posts sorted by topic?

I look forward to the next five “Quarter wit, Quarter Wisdom” posts I have open!

Avi