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inequalities

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inequalities

by RumpelThickSkin » Tue Jul 13, 2010 1:24 am
If x ≠ 0, is
X^2 < 1?
____
/x/


(1) x < 1
(2) x > −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.

[spoiler]OA is C but I think it is A. I don't we need (2) to prove this[/spoiler]. Please explain guys!
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Source: — Data Sufficiency |
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by kvcpk » Tue Jul 13, 2010 1:48 am
If x ≠ 0, is
X^2 / |x|< 1?
(1) x < 1

put x=-1
1/1 <1.. NO
Put x=1/2
1/4/1/2 = 1/2 <1..YES

INSUFF

(2) x > −1
put x=1
1/1 <1.. NO
put x=1/2
1/4/1/2 = 1/2 <1..YES

INSUFF

Combining:

x^2 <|x|
x^2-|x|<0
if x<0, then
x^2+x <0
x(x+1)<0
we know that x<0
There fore x+1 should be greater than 0 for x(x+1) to be <0
hence x+1>0 -> x>-1

If x>0, then
x^2-x<0
x(x-1)<0
We know that x>0
hence x-1<0 for x(x-1) to be <0
hence x<1

pick C
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by gmatruler » Tue Jul 13, 2010 4:54 am
RumpelThickSkin wrote:If x ≠ 0, is
X^2 < 1?

(1) x < 1
(2) x > −1
x^2 would equal 1 if x were 1 or -1. For x^2 to be less than 1, x has to be between 1 and -1. So this question is really asking: "Is x between 1 and -1?"

In the 1st statement, x could be between 1 and -1, but it could also be -10. In the 2nd statement x could be between 1 and -1 but it could also be 10. Only by putting the statements together can we be sure that x is between 1 and -1. That is why C is the answer
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