Solve the following linear inequations in R

Given

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

i. x + 5 > 0 and x – 5 > 0

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

⇒ x > –5 and x > 5

∴ x ∈ (–5, ∞) ∩ (5, ∞)

However, (–5, ∞) ∩ (5, ∞) = (5, ∞)

Hence, x ∈ (5, ∞)

ii. x + 5 < 0 and x – 5 < 0

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

⇒ x < –5 and x < 5

∴ x ∈ (–∞, –5) ∩ (–∞, 5)

However, (–∞, –5) ∩ (–∞, 5) = (–∞, –5)

Hence, x ∈ (–∞, –5)

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

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