Solve the following linear inequations in R

Question

Philoid · Accepted Answer

Given

For this inequation to be true, there are two possible cases.

i. x – 5 > 0 and x – 2 > 0

⇒ x – 5 + 5 > 0 + 5 and x – 2 + 2 > 0 + 2

⇒ x > 5 and x > 2

∴ x ∈ (5, ∞) ∩ (2, ∞)

However, (5, ∞) ∩ (2, ∞) = (5, ∞)

Hence, x ∈ (5, ∞)

ii. x – 5 < 0 and x – 2 < 0

⇒ x – 5 + 5 < 0 + 5 and x – 2 + 2 < 0 + 2

⇒ x < 5 and x < 2

∴ x ∈ (–∞, 5) ∩ (–∞, 2)

However, (–∞, 5) ∩ (–∞, 2) = (–∞, 2)

Hence, x ∈ (–∞, 2)

Thus, the solution of the given inequation is (–∞, 2) ∪ (5, ∞).