Solve the following linear inequations in R
Given
For this inequation to be true, there are two possible cases.
i. x > 0 and 4 – x < 0
⇒ x > 0 and 4 – x – 4 < 0 – 4
⇒ x > 0 and –x < –4
⇒ x > 0 and x > 4
∴ x ∈ (0, ∞) ∩ (4, ∞)
However, (0, ∞) ∩ (4, ∞) = (4, ∞)
Hence, x ∈ (4, ∞)
ii. x < 0 and 4 – x > 0
⇒ x < 0 and 4 – x – 4 > 0 – 4
⇒ x < 0 and –x > –4
⇒ x < 0 and x < 4
∴ x ∈ (–∞, 0) ∩ (–∞, 4)
However, (–∞, 0) ∩ (–∞, 4) = (–∞, 0)
Hence, x ∈ (–∞, 0)
Thus, the solution of the given inequation is (–∞, 0) ∪ (4, ∞).