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

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

• 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

• 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).

• 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).