Hey rb90:
Selango makes a great explanation here, so let me point it out specifically:
With inequalities and absolute values, you have two inequalities to satisfy each time. Either the term inside the absolute value must be greater than the value or it has to be less than the negative of the value:
Example:
|x| > 10
Means that either:
x > 10 OR x < -10
In pretty much every situation with absolute values and inequalities it's best to set the problem up as two individual inequalities to keep your calculations straight.
Here, your job is to find a value of x that works for all values between -5 and 3. A couple clues here:
1) It's not symmetrical (-5 and 3), so you can rule out the choices that have a straight |x| in them because there must be some addition/subtraction to provide diversity of the -/+ values of x.
2) The end values are important - it's likely in a GMAT question like this that the middle values (-1, 0, 1) of this range will be satisfied by the majority of answer choices, but the end values should tip you off. You'll want to test the remaining choices to see how you can get:
3, but not 4
-5 as well as -4
How does that work? Adding 1 should do it - that keeps the positive value of x down (because you'll add to it) and allows you to maximize the absolute value on the negative side (because adding 1 will bring you back toward 0). With that strategy in mind, you can then look at choice E:
|x+1| </= 4
That satisfies -5 (add one and it equals -4) and 3 (add one and it equals 4), so answer choice E is correct.
The biggest key with absolute value and inequalities is to recognize that each "absolute inequality" gives you two inequalities. Treat it as two problems and you'll do well.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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