Question

Prove that if 1 < x < 4 and 1 < y < 4, then |x − y| < 3.

Answer 1

Given: 1 < x < 4 and 1 < y < 4

To prove:

Proof:

Consider the following cases.

Case 1: x < y

As 1 < x < 4 and 1 < y < 4

⇒ 1 < x < y < 4

⇒ 1 − x < x − x < y − x < 4 − x (As x is a positive number)

⇒ 1 − x < 0 < y − x < 4 − x …(1)

Consider,

=

< 4 − x (From (1))

< 3 (As 1 < x < 4)

Case 2: x > y

As 1 < x < 4 and 1 < y < 4

⇒ 4 > x > y > 1

⇒ 4 − y > x − y > y − y > 1 − y (As y is a positive number)

⇒ 4 − y > x − y > 0 > 1 − y …(2)

Consider,

< 4 − y

< 3 (As 1 < y < 4)

Thus, if 1 < x < 4 and 1 < y < 4, then < 3