Algebra Question Bank, Q7

[LauraK659]

In MGMAT Algebra Question Bank Q7, the prompt is "Is |x| < 1 ?" with first prompt:

(1) |x + 1| = 2|x – 1|

Can you please provide some guidance on how to solve complex inequalities such as this? The explanation states that this should be tested if x<-1 (both values in bars are negative), if -1<x<1 (only one value in bars is negative), or x>1 (both values are zero or positive). However I don't have a great understanding on how to quickly apply this methodology to the next double sided absolute inequality I would see.

Thanks in advance,
Laura

[Sage Pearce-Higgins]

It seems that there are several issues here. First of all, rephrasing the question: "Is |x| < 1 ?". Think about using the simplest method available, rather than the "official" or "correct" one. Just as you learn to "translate" words into algebra, you can do the same in reverse: "is the absolute value of x less than 1?". "Absolute value" just means "distance from zero on the number line". You can even draw out a number line to check this; the question becomes "Is -1<x<1?" (i.e. "is x between -1 and 1?").

For dealing with double sided absolute values, remember that those absolute value lines simply knock off a minus sign if there is one and leave the expression alone if it's already positive. So, |x + 1| is equal to x + 1 or -(x + 1). This means that solving the equation |x + 1| = 2|x – 1| has 4 strands to it:
(x + 1) = 2(x – 1), that's both sides positive
- (x + 1) = 2(x – 1), that's negative / positive
(x + 1) = 2 ( - (x – 1)), that's positive / negative
- (x + 1) = 2( - (x – 1)), that's both sides negative
Try solving these equations and see what happens (you'll get some repeated results, why?). Look at p158 of the Algebra strategy guide for more examples.

As for solving absolute values on both sides of an inequality, you use the same principles. Things can get pretty tricky. but I wouldn't worry about that if I were you, as you're very unlikely to see such a question in GMAT.