Эффективные уроки GMAT. Inequalities with absolute value

To solve an inequality with the absolute value you need to remember two simple rules:


1.    Is |x - 1| < 1?
(1) (x - 1)2 d 1
(2) x2 - 1 > 0

(A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient
(B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient

In fact, we are asked if -1< x-1< 1 or if 0< x< 2.
Statement (1) is equivalent to |x - 1| d 1 or 0d xd 2 . If x = 0 or x = 2, statement (1) holds but the principal statement is not true. Statement (2) is not sufficient either. If x = 1.5 the principal statement holds but if x is big, say 5, the principal statement is not true. Although Statement (2) means that x cannot be 0 it does not exclude the possibility of x being 2. Thus, Statement (1) and Statement (2) combined are not sufficient to answer the question. The answer is E.
2.    Is |x - 5| > 4 ?

(1) x is an integer
(2) x2 < 1

(A) Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient
(B) Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient

This question becomes incredibly easy if we rewrite it:
|x-5|>4 is the same as x-5>4 or x-5<-4, so the question is if x9.
Statement (1) doesn't help at all. Statement (2) is sufficient for it means that x is confined between -1 and 1 and thus its true that x<1. The answer is B.
3.    How many integers X are there so that |X - 3.5| < 2?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
This inequality can be rewritten as -2Resume:
This lesson shows some typical problems, in which inequalities with the absolute value appear, and the most effective ways to solve them in 2 minutes or even faster and add some points to your GMAT total score.

Material prepared by Ksenia Zueva, GMAT consultant at MBA Strategy