Question

Solve the following linear inequations in R

x/(x-5) > 1/2

\(\frac{x}{x-5}\) > \(\frac{1}{2}\)

Answer 1

Given,

x/(x-5) > 1/2

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, ∞).