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, ∞).

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