Inequality Testing Ranges

[kouranjelika]

Hi Ron,

I used your approach in solving quadratic inequalities via plotting the solutions on a number line and then testing cases in the ranges between them. But I am either making a mistake or it isn't working on this problem:

a^3-a^2-2a<0
I factor out an a and then solve the quadratic:
a(a^2 - a - 2) <0
a(a-2)(a+1)<0
So from here I plotted the three possible solutions:
a=0 a=2 a=-1 and started testing the regions for validity.
When I got to the upper range, I tested between 0 and 2 and was super confused. Because for 1/2 it works, for 1 it doesn't. But why? I mean numerically I get why, but I don't understand why the quadratic method did not work?

This question is actually a DS one and the question itself asks if a<0?
So if I wouldn't have tested the 1/2 and just took any number in the range between the two solutions (0 & 2), which would typically be 1 because it's a nice number and I wouldn't see any reason to test half, I would find the statement to be in fact valid when it is not.

Can you explain why the quadratic did not work and if you recommend applying the quadratic to these generally at all? I know there is no cookie-cutter approach and we must stay dynamic, but I definitely NEVER want to use an approach which in some cases leads me to an incorrect answer.

Thanks!
Anjelika

[RonPurewal]

You must have made an arithmetic error somewhere; both 1 and 1/2 make the inequality true.

If you use the factored form, then, for both values, a(a-2)(a+1) is (positive)(negative)(positive).

If you go back and plug into the original inequality (more advisable, as it's immune to any errors one might make in the factoring process), then plugging in 1 and 1/2, respectively, gives
1 - 1 - 2 < 0 --> -2 < 0 (true);
1/8 - 1/4 - 1 < 0 (clearly true; no need to bother finding the exact value, since 1/8 - 1/4 is already negative).

[RonPurewal]

This method will be 100% reliable for any inequality involving polynomials (not just quadratics). There are no instances in which it will fail"”except, of course, instances of human error in the calculations themselves.

Think about it for a sec: For this method NOT to work, the values of the function would somehow have to change signs without going through zero!
E.g., you're saying that you get different signs by plugging in 1/2 and 1. There's no way that could be true"”if these values had different signs, then, somewhere between x = 1/2 and x = 1, the polynomial would have to be zero.

[kouranjelika]

Hi Ron,

Thanks for this. I don't know how I calculated (I did actually plug into the original inequality but still got something random).

Super. Happy to know I can def rely on this and yea, it totally makes sense.

Thank you!

P.S. I know in the study hall last week you mentioned that Gmat Prep has 900 questions. Which questions were you referring to? Because I thought we all got a standard pack of 90 and they were all the same. Where do you get the 900?

Thanks!!

[RonPurewal]

P.S. I know in the study hall last week you mentioned that Gmat Prep has 900 questions. Which questions were you referring to? Because I thought we all got a standard pack of 90 and they were all the same. Where do you get the 900?

Thanks!!

That's the approximate number of questions that are in the pool for the 2 free practice tests.

There are archives of these floating around the internet"”if you poke around, you can find some. I don't think anyone has a complete set.

[kouranjelika]

Oh I see!

So I can, in fact, post questions on the forums from the practice tests? Good to know! Thanks!

[kouranjelika]

Also, Ron, do you know if we buy the extra exam pack, are the questions still coming from this same 900 pool or are they fresh?

[RonPurewal]

Oh I see!

So I can, in fact, post questions on the forums from the practice tests? Good to know! Thanks!

Anything from the free GMAT PREP material is fair game, because it's free material. (The GMAT PREP math and verbal folders"”the latter of which is the most active folder on the forum"”consist entirely of problems from the free practice tests.)

We don't allow problems from the paid GMAT PREP exam packs, for the same reason why we don't allow OG problems.

[RonPurewal]

Also, Ron, do you know if we buy the extra exam pack, are the questions still coming from this same 900 pool or are they fresh?

The paid exam pack shouldn't overlap with the free material. It's a paid product, after all, while the free material is free.
If there were overlap, I would cry foul.

Interestingly, while some of the exam pack problems seem to be genuinely new, others have been found in even older GMAT exams (pre-10th edition OG).
From these observations"”as well as from the fact that the actual problems from one's official GMAT exam are never released"”we can conclude that GMAC very seldom creates new problems for the exam.

[kouranjelika]

Whatever you just said should be a critical reasoning problem.

[RonPurewal]

Perhaps.