Rules behind absolute values

[NL]

Hi Ron,

I’d like to know why members of group (1) work and those of group (2) don’t. What are rules behind them? Thank you.

(1) Something like these work:
|a| = 3a-8
|1/3a| = 1/2a -1
|2a| = 3a-8
|10a| = 100a-1/100 etc.

(2) Something like these don’t work:
|1/2a| = 1/3a-1
|1/2a|= -1/3a-1
|3a| =2a-8
|100a|= 10a- 1/100

[RonPurewal]

I can't give you "rules"; I don't know a whole lot of those.

One insight, though, comes from thinking about the graphs of these functions in the x-y plane.
The graph of y = |x| is two half-lines at 45º angles to the coordinate axes, through the upper left and upper right quadrants.
If there's a coefficient, then the slopes of the lines are different, but the idea is the same. E.g., the slopes in y = |2x| are a little steeper.

Thinking about something like |x| = 3x - 8:
The line y = 3x - 8 has a y-intercept of -8. Clearly it's not going to intersect y = |x| to the left of the y-axis, since it's negative while |x| is positive.
To the right, it has to "catch up" vertically with the graph of |x| (since it starts at -8 while |x| starts at 0). Because its slope is greater, this will happen, and there will be an intersection point.

Now, thinking about something like |3x| = x - 8:
|3x| has a steeper slope than x - 8, and it starts out with a higher y-intercept. So, x - 8 will never "catch up" to it. (In fact, the closest the two graphs will ever get is when x = 0.)

I'm sure that's not the only approach, but it's an approach.
Possible disadvantage: It does demand a greater-than-usual intuition about graphs of absolute value functions.
Possible advantage: It doesn't require much work.

[RonPurewal]

By the way, the following must be noted:

* All of the following is well beyond the scope of what GMAC would ever actually test on the exam.

* By thinking about these things, you may compromise your performance on the test.
All of these issues can be settled routinely"”and quickly"”by just doing the work; there's no practical reason to think about any of this stuff.
I find it somewhat interesting"”but I also realize that it is not anything I would ever think about while taking a test! Make sure you realize the same thing. Thinking about things like these may be enjoyable recreation, but it will NOT help you do better on the GMAT.

[NL]

One insight, though, comes from thinking about the graphs of these functions in the x-y plane.

Hey, you’re a real expert, Ron :))

So, if a grap is steeper, it should be "deeper" in order to cut (intercept) the not-steeper graph at at least one point.
It also explains why|3a|=2a-8 doesn’t work, but |3a|=2a+8 work, or it doesn’t matter with -|3a|=2a+8
Interesting!

Why do you think about graphs? (maybe your experience told you). What if "=" is replaced by ">" (stop! It’s not what gmat will test)

By thinking about these things, you may compromise your performance on the test.

Thank you for the reminding. Sometimes I had a thought of giving up when I couldn't find a short cut.

Thinking about things like these may be enjoyable recreation, but it will NOT help you do better on the GMAT.

Don’t you know practicing Kung-fu helped Bruce Lee win a chachacha championship?

But you maybe right: doing this thing too much will lead me to the wrong direction. (Ironically, I never could focus on doing questions as much as doing these)

But you may not be right: The effect of emotion is still a mystery. It may not help my test directly, but somehow keep me stay with the test.
Hem hem...

[RonPurewal]

Why do you think about graphs?

The real reason? I didn't have a pen and paper handy. Large-scale graphing, unlike algebra, I can do mentally. (Probably couldn't find exact intersection points, but that's not necessary here.)

But you maybe right: doing this thing too much will lead me to the wrong direction. (Ironically, I never could focus on doing questions as much as doing these)

This isn't what I meant. I meant that "mental clutter" will not serve you well.

Here, you have a technique that's perfectly straightforward already"”i.e., just solving the equations"”and that doesn't have pitfalls or exceptions. Why mess with success?

[NL]

"mental clutter" will not serve you well.

This words become my mantra every time I catch my mind wandering around :)

[NL]

The real reason? I didn't have a pen and paper handy. Large-scale graphing, unlike algebra, I can do mentally.

Go watch Bruce Lee-Ping pong on youtube! Is this how your brain interacted with questions on the test that you got 800? Jealous!

"mental clutter" will not serve you well! Wandering again, :(

[RonPurewal]

Hmm.