Inequalities and Mods

Mission2012 wrote:HI All,

I am really struggling with the questions pertaining to inequalities and specifically when they includes mods.

Could anyone suggest a good source to develop concepts and to practice question on the same.

Regards,

Dear Mission,
I'm happy to respond.

I have a question about your terminology. You use the term "mods" ---- I assume you are not referring to modular arithmetic, which is a topic well beyond the GMAT. Are you referring to what is properly known as "absolute value" --- i.e. |-3| = +3? If so, I can make some recommendations. If you mean something else by "mods", please clarify.

Mike

Mission2012 wrote:Hi Mike,

Thanks for your reply. I meant absolute value (modulus). Could you please suggest how I can work on this weakness.

Regards,

Dear Mission2012,
You are not likely to see many problems such as this on the real test. The OG has almost nothing on this topic. For more information, see this blog:
https://magoosh.com/gmat/2012/gmat-math- ... te-values/

Suppose we have an equation in absolute values, and there's an inequality.
|equation| > 3

Then, this means the acceptable values could be
equation > +3 or equation < -3

Instead, if the original inequality were:
|equation| < 3

Then, this means the acceptable values would be
-3 < equation < +3

Either way, to find this, we always solve for the equations: equation = +3 and equation = -3.

The solutions to those two equations will divide the number line into subsets or segments, and then you must test each segment individually, if you can't figure out by logic which segments work or don't work in the original inequality.

That's how we would solve if we had a full-blown equation, say a quadratic, in an absolute value inequality. A problem such as that, though, would be in at the very upper limit of the difficult questions the GMAT Quant will give you --- you would have to be acing everything else to get a question such as that.

More often than not, for example in OG PS #143, when the GMAT is asking about absolute value inequalities, it is not asking you to do a lot of algebra --- instead, it is testing your intuitive understanding of the distance definition of the absolute value. I discuss this in a little more depth in the link above, but here's an overview.

Fundamentally, subtraction is about distance on the number line. That's a big idea in and of itself.

|x| = the distance x is from zero
|x - 5| = the distance x is from the point +5 on the number line
|x + 3| = the distance x is from the point -3 on the number line

Thus the absolute value inequality
|x - 7| < 2
is not something that needs any algebra to solve. Just think about the logic. We are looking for numbers, points on the number line, that are a distance of less than two from the number +7. Therefore, the points that satisfy this must be in the region
7 - 2 < x < 7 + 2
5 < x < 9

More often than not, with absolute values, as with many other operations in math, the GMAT is not trying to get you to do a ton of calculations. Rather, it is almost always looking for something quite elegant, something you solve primarily with logical insight rather than with brute force calculations. See:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/

Does all this make sense?
Mike

Hi Mission2012,

Absolute Value questions aren't too frequent of an issue on the actual GMAT (you might see 1-2 in total). Sometimes these questions involve inequalities, sometimes they don't.

Assuming a variable is involved, you can normally AVOID doing algebra and just FIND the values that fit the Absolute Value/inequality.

For example:

| X + 2 | > 5

Don't over-think this, just figure out what the possibilities are. Since it's an absolute value, there should be some positive possibilities and some negative ones. With a bit of "brute force", you should be able to figure out the "borders."

Here X CAN'T = 3 because 3 + 2 is NOT > 5, but X COULD be > 3
Also, X CAN'T = -7 because |-7 + 2| = |-5| = 5 exactly, but X COULD be < - 7

Most GMAT questions can be solved with a variety of approaches, so there's some advantage to being flexible with your thinking. Sometimes the "math approach" takes too long and the "fast way" to get the answer is just a mix of TESTing values and "brute force."

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