Question

9. Rezolvă, în mulțimea numerelor reale, inecuațiile: a) (x-2)(x - 3) ≤ x(x - 1); b) 3x(2x - 5) ≥ x(6x-7); c) 2x(1-x) + 4x(2x + 1) + 3x(5 - 2x) ≤ 0; d) (2x + 1)(12x-5) ≤ (4x-3) (6x + 4); e) (x + 2)(x+3) + (3x + 1)(2x + 1) ≥ 7(x + 1)(x + 2).​

Answer 1

Răspuns:

Explicație pas cu pas:

a) (x-2)(x - 3) ≤ x(x - 1) <=> x²-5x+6 ≤ x²-x => 6 ≤ 4x  I:2 =>

3 ≤ 2x => x ≥ 3/2 => x ∈ [3/2 ; +∞)

b) 3x(2x - 5) ≥ x(6x-7) <=> 6x²-15x ≥ 6x²-7x => 0 ≥ 8x  =>

x ≤ 0 => x ∈ (-∞ ; 0]

c) 2x(1-x) + 4x(2x + 1) + 3x(5 - 2x) ≤ 0 <=>

2x-2x²+8x²+4x+15x-6x² ≤ 0 <=>

21x ≤ 0 => x ≤ 0 => x ∈ (-∞ ; 0]

d) (2x + 1)(12x-5) ≤ (4x-3) (6x + 4) <=>

24x²+2x-5 ≤ 24x²-4x-12 <=>

6x ≤ -7 => x ≤ -7/6 => x ∈ (-∞ ; -7/6]

e) (x + 2)(x+3) + (3x + 1)(2x + 1) ≥ 7(x + 1)(x + 2) <=>

x²+5x+6 +6x²+5x+1 ≤ 7x²+21x+14 <=>

7-14 ≤ 21x-10x => -7 ≤ 11x => x ≥ -7/11 => x ∈ [-7/11 ; +∞)