Inequalities with Abs. Values in Denominator

[JacquelineD347]

I reviewed Ch. 8 (Ineq and Abs Values) in Foundations of Math, but unfortunately it does not touch upon how to solve inequalities with absolute values in the denominator.

Two examples that I'd love to be walked through step by step are the following:

A) x/|x|<x

B) 2/|x+1|<4

Might you also know of a Strategy Guide that goes into further detail with example sets?

Thank you!

[RonPurewal]

I reviewed Ch. 8 (Ineq and Abs Values) in Foundations of Math, but unfortunately it does not touch upon how to solve inequalities with absolute values in the denominator.

if you're looking for an algebraic solution then this is hardly a surprise, because no such animal exists.

think of how you would solve something like (x + 2)(x - 1) < 0:
• find where the left side EQUALS zero (here, that would be x = -2 and x = 1)
• those values divide the number line up into smaller intervals (to the left of -2... between -2 and 1... to the right of 1).
• TEST those intervals. i.e., just take one number from each interval, throw it into the equation, and see whether it works. (here, only the middle interval will work.)

you can take a similar approach here.

--

below, i will write some outlines that will allow you to start investigating these problems.

i'm not going to provide complete solutions, since that would essentially defeat the purpose—which is for YOU to do the critical thinking, in order to develop the relevant aspects of your intuition.

[RonPurewal]

A) x/|x|<x

first, you should investigate the left-hand side by itself.
• take a bunch of numbers—some positive, others negative. (you can't use zero here, since 0/0 is not a thing.)
• plug them into x / |x|.
• this will be the world's easiest pattern to identify. identify what happens (and take a moment to understand WHY it happens).

NOW...
• think about when x / |x| might equal x. (you can't 'solve' this—but it's not hard to come up with the values, since there are so few possibilities for what the left-hand side can be.)
• divide the number line up, using these values and also 0 (where the expression isn't defined). if you're doing everything right, the number line should be divided into 4 regions.
• test each of the 4 regions.
• your answer consists of the intervals that work.

[RonPurewal]

B) 2/|x+1|<4

––– THIS IS VERY UNLIKELY TO HAVE ANYTHING TO DO WITH THE GMAT EXAM –––

read that ^^ several times. then read it again. DO NOT spend too much effort on this problem; it is not the kind of thing that shows up on this exam.

when absolute values show up on this exam, they tend to show up in ways that...
...are highly conceptual,
OR
...admit easy pattern recognition.

this problem is neither of those things.
rather, this problem is the sort of thing that would come at the end of a high-school homework assignment—in other words, exactly the sort of thing that DOESN'T show up on this exam.

but:

FIRST you want to find where 2 / |x + 1| EQUALS 4.
• the left-hand side could be one of two things: it could be 2 / (x+1), or it could be 2 / (-x - 1).
• set each of these equal to 4.
• solve.
• check in the original equation (with the absolute value) to make sure it works.

now solve the inequality:
• you have the solutions to 2 / |x + 1| = 4, which you found above. (there should be two of them.)
• the fraction isn't defined at x = -1, so you have to consider that point too.
• you should now have 3 numbers to put on the number line.
• using those numbers, divide the number line into 4 regions.
• test them.

––– THIS IS VERY UNLIKELY TO HAVE ANYTHING TO DO WITH THE GMAT EXAM –––

[RonPurewal]

Might you also know of a Strategy Guide that goes into further detail with example sets?

if you find 'example sets' similar to the second problem here, they will be a waste of your time (as far as the GMAT exam is concerned, at least).

the point here, more than anything else, is pattern recognition.

the first problem here is more like the things that show up on this exam.
but, ironically, that is exactly the reason why it's NOT important to find tons and tons of problems that look like it.

consider:

• to solve the first problem, the only thing you need to know is the basic definition of absolute value (so that you can plug numbers into the left-hand side and investigate it).

• in that problem, no other memorized facts are necessary (...and even if you have memorized a hundred billion thousand different things about absolute values, those things really won't help you here).

• to solve the problem you just have to investigate what the expressions do, and then think a bit about when they might be equal or not equal.

THIS is the sort of thing that shows up on the gmat exam: problems that you can solve with elementary knowledge + a spirit of investigation.

[JacquelineD347]

Thank you for clarifying!

[tim]

[RonPurewal]

Thank you for clarifying!

sure.