If x is less than 2, which of the following statements are always true

Question

If x is less than 2, which of the following statements are always true?

I. x is negative
II. x is positive.
III. 2x is greater than or equal to x
IV. x^2 is greater than or equal to x.

A. III only
B. IV only
C. I and III only
D. I, III, and IV only
E. none of the statements

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Rohit2015 · Answer

I. x is negative-------> x may be 1--Discard
II. x is positive.--------> x may be -1--Discard
III. 2x is greater than or equal to x-------> x may be -2---Discard
IV. x^2 is greater than or equal to x.-----> x may be 0.5---Discard

none is correct
Ans E

GMATWhizTeam · Answer

Solution 
 • It is given that x < 2.
• No further information is given so, lets directly check each statement and see which one is always true.
•  I. x is negative 
 o Let’s consider any 2 possible values of x which are less than 2
  Say, x = -1, in this case x is negative.
 Or, x = 1, in this case x is positive    
Hence, statement I may be true in some cases, but it is NOT always true.
 •  II. x is positive. 
 o Let’s consider any 2 possible values of x which are less than 2
  Say,  x = -1,  in this case x is negative.
 Or,  x = 1 , in this case x is positive     
Hence, statement II may be true in some cases, but it is NOT always true.

•  III. 2x is greater than or equal to x 
 o Let’s consider any 2 possible values of x which are less than 2
  Say,  x = -1⟹ 2x = -2   which is less than -1(or x).
 Or,  x = 1, ⟹2x = 2   which is more than 1(or x).    
Hence, statement III may be true in some cases, but it is NOT always true.

•  IV.  x^2   is greater than or equal to x. 
 o Let’s consider any 2 possible values of x which are less than 2
  Say,  x = \frac{1}{2} ⟹x^2 = \frac{1}{4}  which is less than  \frac{1}{2}  (or x)
 Or,  x = 1, ⟹ x^2 = 1  which is equal to 1(or x).    
Hence, statement IV may be true in some cases, but it is NOT always true.
Thus, the correct answer is  Option E.

KarishmaB · Answer

Just think of the number line. x is to the left of 2.

.................-1........... 0 ............1........... 2 .........................
<-----------------------------------------------

I. x is negative
Numbers to the right of 0 are not negative. Not true

II. x is positive.
Numbers to the left of 0 are not positive. Not true

III. 2x is greater than or equal to x
When you multiply negative integers by 2, they will become every more negative. So they will be smaller. Not true

IV. x^2 is greater than or equal to x.
Numbers between 0 to 1, when squared, are smaller. 
 (1/2)^2 < 1/2 
Not true

Answer (E)

Doer01 · Answer

Hi, IMO E. x<2 does not tells us the lower limit of x. I. x COULD be negative II. x COULD be positive III. 2x>x so long as x<0 IV. x^2 > x only if x<1. So none of the statement is MUST BE true.

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If x is less than 2, which of the following statements are always true

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If x is less than 2, which of the following statements are always true

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