Question about inequalities and absolute value?

Question

Hi guys,

I have a question that involves "opening" an absolute value:

Hypothetical question: If \(|x| > -8\), logic dictates that x must be lower than -8 or larger than 8. But when I do the actual calculation, I get the opposite result:
\(|x| > -8\)
\(x > -8 and x < -(-8)\)
\(-8 < x < 8\)

Can someone comment on what I'm doing incorrectly? dabral?

EMPOWERgmatRichC · Answer

Hi bionication,

Did you come up with this 'hypothetical' on your own? I ask because from a basic "concept" standpoint, |X| is ALWAYS greater than or equal to 0. It can NEVER be negative. So stating |X| > -8 is redundant.

However, if you were dealing with something like |Y| > 7....

Then the possible values of Y would be Y > 7 or Y < -7.

GMAT assassins aren't born, they're made,
Rich

bionication · Answer

Hey  EMPOWERgmatRichC ,

Thanks for the explanation. The question I posed was in response to the following question from MGMAT's Advanced Guide (Set 8, Q80): An (x, y) coordinate pair is to be chosen at random from the xy-plane. What is the probability that y ≥ |x|?

I ended up solving the question using coordinate geometry since I couldn't approach it algebraically. Analogous to how |x| = 3 is x = +/-3, I believed that y ≥ |x| could be re-written as y ≥ x and x ≥ -y. But y obviously cannot be negative, so there must be some inherent restriction to correctly "opening" an absolute value. I haven't come across such a question and would like to learn an algebraic approach to inequalities like the one above. Your help is appreciated.

Thanks!

EgmatQuantExpert · Answer

Dear bionication The red part in your post is erroneous. And so is the blue part. Here's how you should think through an inequality like: |x| > -8 1. The Visual Method |x| represents the distance of a number x from the point 0 on the number line. This means that the inequality |x| > - 8 represents those numbers that are at a distance greater than -8 units from 0 on the number line. Now, note that 'distance' is always non-negative. For example, the points -3 and +3 are both said to be at a distance of 3 units from the point 0 on the number line. So, what is the minimum possible distance between 2 points on the number line? The answer is: ZERO (when 2 points coincide on the number line). So, what are the numbers that are at a distance greater than -8 units from 0? The answer is: ALL NUMBERS are at a distance greater than -8 units from 0 (because the distance of any number from 0 will be greater than or equal to 0) So, Negative Infinity < x < Positive Infinity is the correct answer for |x| > -8 2. The Algebraic Method The given inequality is |x| > - 8 Case 1: x is non-negative So, |x| = x Thus, given inequality becomes: x > - 8 . . . (1) But for what values of x is x non-negative in the first place? For x > = 0 . . . (2) So, the values of x that satisfy Case 1 are obtained by the overlap of inequalities (1) and (2), which is x > = 0 So, Range of x that satisfies Case 1: x > = 0 Case 2: x is negative So, |x| = -x Thus, given inequality becomes: -x > - 8 Multiplying both sides of the inequality with -1 will flip the sign of inequality: x < 8 . . . (1) But for what values of x is x negative in the first place? For x < 0 . . . (2) So, the values of x that satisfy Case 2 are obtained by the overlap of inequalities (1) and (2), which is x < 0 So, Range of x that satisfies Case 2: x < 0 (Note: you forgot to consider Inequality (2) in both cases 1 and 2. This is why, you got only x > -8 from Case 1 and likewise, only x < 8 from Case 2. I hope you now realize how important it is to also consider the set of values of x for which a particular case holds true in the first place

) Thus, we see that the given inequality |x| > -8 is satisfied by x > = 0 or by x < 0. Basically, this inequality is satisfied by ALL POSSIBLE REAL values of x from Negative Infinity to Positive Infinity. I hope this explanation was useful for you!

Best Regards Japinder

EMPOWERgmatRichC · Answer

Hi bionication,

It looks like this post must have gotten 'buried', but I'm happy to offer some additional suggestions (assuming that you're still around to read them).

Taking an "all-math" approach to the Quant section of the GMAT is a dubious move, since so many of the questions can be solved by tactical approaches, TESTing THE ANSWERS, Number Properties, pattern-matching, estimation, etc.

Here, I'll bet that you can draw a picture and TEST VALUES to come up with the solution rather easily. Here's how to do it...

We're dealing with a co-ordinate plane and we're looking for the part of that plane in which Y ≥ |X|...

Let's start simple (draw a graph):

IF...X=0, what values of Y on the graph would "fit"? Y ≥ 0.
"Shade" that in on the graph.

What if X = 1 or X = -1, what values of Y on the graph would "fit? Y ≥ 1.
"Shade" THAT in on the graph (for both X = 1 and X = -1).

How about when X = 2 or X = -2....?
Now "shade" that in too.

Notice a pattern forming? What fraction of the graph are we dealing with? THAT fraction is the answer to the question. No complex calculations required.

GMAT assassins aren't born, they're made,
Rich

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Question about inequalities and absolute value?

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