Solve the following linear inequations in R

Given

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

i. x + 7 > 0 and x + 3 < 0

⇒ x + 7 – 7 > 0 – 7 and x + 3 – 3 < 0 – 3

⇒ x > –7 and x < –3

∴ x ∈ (–7, ∞) ∩ (–∞, –3)

However, (–7, ∞) ∩ (–∞, –3) = (–7, –3)

Hence, x ∈ (–7, –3)

ii. x + 7 < 0 and x + 3 > 0

⇒ x + 7 – 7 < 0 – 7 and x + 3 – 3 > 0 – 3

⇒ x < –7 and x > –3

∴ x ∈ (–∞, –7) ∩ (–3, ∞)

However, (–∞, –7) ∩ (–3, ∞) = ∅

Hence, this case is not possible.

Thus, the solution of the given inequation is (–7, –3).

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