If x = −2, which of the following is(are) false?

Solution :

As we know | x | = x if x ≥ 0
= −x if x < 0

Given x = −2 ⇒ |x| = |−2| = 2

Considering statement I :-

|x| = −|x| + |2x|

LHS = |x| = 2

RHS = −|x| + |2x| = −(2) + |−4| = −2 + 4 = 2

Since LHS = RHS ⇒ I is true

Considering statement II:-

(x)−x = (−x)x

LHS = (x)−x = (−2)−(−2) = (−2)2 = 4

RHS = (−x)x = (2)(−2) = 1/4

Since LHS is not equal to RHS ⇒ II is False

Considering statement III:-

(-x)0 = −x/|x|

LHS = {-(-2)}0 = (2)0 = 1

RHS = -x/|x| = −(−2)/(2) = 1

Since LHS = RHS ⇒ III is True

Considering statement IV:-

(1/x)|x| = |x|x

LHS = (1/x)|x| = (−1/2)(2) = 1/4

RHS = |x|x = (2)(−2) = 1/4

Since LHS = RHS ⇒ IV is also True.

Therefore, only statement II is false

Answer: E.