Question

Solve the following linear inequations in R 

(5x+8)/(4-x) < 2

\(\frac{5x+8}{4-x}\) < 2

Answer 1

Given,

(5x+8)/(4-x) < 2

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, ∞).