Disequazioni 🚨🚨🚨🚨🚨🚨🚨🚨⬅️⬅️⬅️⬅️

Risolviamo separatamente le tre disequazione

1^ disequazione

(x - 3)/(x^2 - 2·x + 1) + 2·x/(x^2 - 3·x + 2) ≤ 3/(x - 2)

(x - 3)/(x - 1)^2 + 2·x/((x - 1)·(x - 2)) - 3/(x - 2) ≤ 0

((x - 3)·(x - 2) + 2·x·(x - 1) - 3·(x - 1)^2)/((x - 1)^2·(x - 2)) ≤ 0

((x^2 - 5·x + 6) + (2·x^2 - 2·x) - (3·x^2 - 6·x + 3))/((x - 1)^2·(x - 2)) ≤ 0

(3 - x)/((x - 1)^2·(x - 2)) ≤ 0

(x ≠ 1 ∧ x < 2) ∨ x ≥ 3

2^ disequazione

x - 1/3·(x - 2)·(x + 3) - 1/3·(4 - x)·(x + 2) ≤ 1/4·(2 - x)

12·x - 4·(x - 2)·(x + 3) - 4·(4 - x)·(x + 2) ≤ 3·(2 - x)

12·x - (4·x^2 + 4·x - 24) - (- 4·x^2 + 8·x + 32) ≤ 6 - 3·x

-8 ≤ 6 - 3·x

x ≤ 14/3

3^ disequazione

(x + 2)/(1/2 - x) + (3 + x)/(x - 1/2) ≤ 3

(x + 2)/(1/2 - x) - (3 + x)/(1/2 - x) - 3 ≤ 0

((x + 2) - (3 + x) - 3·(1/2 - x))/(1/2 - x) ≤ 0

(3·x - 5/2)/(1/2 - x) ≤ 0

(6·x - 5)/(1 - 2·x) ≤ 0

x < 1/2 ∨ x ≥ 5/6

Risolvo quindi:

{(x ≠ 1 ∧ x < 2) ∨ x ≥ 3

{x ≤ 14/3

{x < 1/2 ∨ x ≥ 5/6

ottengo:

[x < 1/2, x ≠ 1 ∧ 5/6 ≤ x < 2, 3 ≤ x ≤ 14/3]