fullymooned wrote:
|x-3| +|x+4| <= |x+8|
How to go about solving this equation for x.
If this equation had one or two absolutes, I am able to solve this, but this one is new.
Dear
fullymooned,
I'm happy to help.
First of all, here's a blog article about GMAT problems with inequalities, including some absolute value inequalities.
https://magoosh.com/gmat/2013/gmat-quant ... qualities/The question you are asking is 100% unlikely to show up on the GMAT. That is just too hard. The GMAT does ask that kind of stuff.
The only efficient way I know how to solve this is graphing it on a graphing calculator or computer. That gives the answer
-1 <= x < = 7
as the solution.
To solve it algebraically, we would have to locate the three points where each absolute value equals zero -- those would be x = {-8, -4, +3}. Then, we would have to divide the real number line into regions based on these division:
---------- (-8) --------- (-4) -----------(+3) --------------
Call those four regions I, II, III, and IV. Then, in each one of those regions, we would have to solve a different algebraic inequality. I will simply demonstrate one solution, the solution in the region III. In region III, between x = -4 and x = +3, we know that the absolute values |x + 4| and |x + 8| will be positive numbers, so those absolute value expressions can be replace with the arguments inside the absolute values unchanged. By contrast, in this region, (x - 3) would be negative, so |x - 3| would have to be replaced with its opposite, -(x - 3) = 3 - x. Sub all these into the inequality above:
(3 - x) + (x + 4) <= (x + 8)
7 <= x + 8
-1 <= x
This is a value that is in the region, so this is part of the final solution. The expression -1 <= x denotes one part of the solution.
That's just a solution for one region, but we would have to do that in all four regions to get the full solution. I absolute guarantee: the GMAT does not expect you to do this kind of algebra. You will never see this on the GMAT.
Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)