Solve the following linear inequations in R

Given

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

i. x – 3 > 0 and x + 6 < 0

⇒ x – 3 + 3 > 0 + 3 and x + 6 – 6 < 0 – 6

⇒ x > 3 and x < –6

∴ x ∈ (3, ∞) ∩ (–∞, –6)

However, (3, ∞) ∩ (–∞, –6) = ∅

Hence, this case is not possible.

ii. x – 3 < 0 and x + 6 > 0

⇒ x – 3 + 3 < 0 + 3 and x + 6 – 6 > 0 – 6

⇒ x < 3 and x > –6

∴ x ∈ (–∞, 3) ∩ (–6, ∞)

However, (–∞, 3) ∩ (–6, ∞) = (–6, 3)

Hence, x ∈ (–6, 3)

Thus, the solution of the given inequation is (–6, 3).

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