Question

if x€R, solve 2x-3>=x+(1-x)/3>2x/5
also represent the solution on number line​

Answer 1

Given info : x ∈ R , where 2x - 3 ≥ x + (1 - x)/3 > 2x/5

solution : case 1 : 2x - 3 ≥ x + (1 - x)/3

⇒6x - 9 ≥ 3x + (1 - x)

⇒6x - 9 ≥ 2x + 1

⇒4x - 10 ≥ 0

⇒x ≥ 5/2 ....(1)

case 2 : x + (1 - x)/3 > 2x/5

⇒5[x + (1 - x)/3] > 2x

⇒5[3x + (1 - x)] > 2x × 3

⇒15x + 5(1 - x) > 6x

⇒15x + 5 - 5x > 6x

⇒7x + 5 > 6x

⇒x + 5 > 0

⇒x > -5 ....(2)

take common values of equations (1) and (2) by putting them in number line.

see figure, it is clear that the solution of given inequality is x ≥ 5/2, i.e., x ∈ [5/2 , ∞ )