Write the set of values of x satisfying the inequations 5x + 2 < 3x +

1.	Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and \(\frac{x+2}{x-1}\) < 4.
Answer» Part I : 5x + 2 < 3x + 8. ⇒ 2x < 6 ⇒ x < 3 Part II : (x+2)/(x-1) < 4. ⇒ \(\frac{x+2}{x-1}\) - 4 < 0 ⇒ \(\frac{(x+2)-4(x-1)}{x-1}\)< 0 ⇒ \(\frac{-3x+6}{x-1}\) < 0 Case I : - 3x + 6 < 0 and x – 1 > 0 ⇒ x > 2 and x > 1 By taking intersection x ∈ (2, ∞) Case II : - 3x + 6 > 0 and x – 1 < 0 ⇒ x < 2 and x < 1 By takin intersection x ∈ (-∞, 1) Taking union of case I and case II, x ∈ (-∞, 1) (2, ∞) We have to take intersection of part I and part II, We have final answer i.e., x ∈ (-∞, 1) (2, 3)

Write the set of values of x satisfying the inequations 5x + 2 < 3x + 8 and \(\frac{x+2}{x-1}\) < 4.

Answer»

Part I : 5x + 2 < 3x + 8.

⇒ 2x < 6

⇒ x < 3

Part II : (x+2)/(x-1) < 4.

⇒ \(\frac{x+2}{x-1}\) - 4 < 0

⇒ \(\frac{(x+2)-4(x-1)}{x-1}\)< 0

⇒ \(\frac{-3x+6}{x-1}\) < 0

Case I : - 3x + 6 < 0 and x – 1 > 0

⇒ x > 2 and x > 1

By taking intersection x ∈ (2, ∞)

Case II : - 3x + 6 > 0 and x – 1 < 0

⇒ x < 2 and x < 1

By takin intersection x ∈ (-∞, 1)

Taking union of case I and case II,

x ∈ (-∞, 1) (2, ∞)

We have to take intersection of part I and part II,

We have final answer i.e.,

x ∈ (-∞, 1) (2, 3)