Răspuns:
1. -4≤3x+2≤4
-4≤3x+2; -6≤3x /:3; x≥-2; x∈[-2; +∞)
3x+2≤4; 3x≤2; x≤[tex]\frac{2}{3}[/tex]; x∈(-∞; [tex]\frac{2}{3}[/tex]
x∈[-2; [tex]\frac{2}{3}[/tex]
2. x²-5x+5≤1
x²-5x+4≤0; x²-x-4x+4≤0; x(x-4)-4(x-1)≤0; (x-1)(x-4)≤0
x-1=0; x₁=1
x-4=0; x₂=4
x₁+x₂=5
3. x²-5x+6≤0
x²-4x-x+4+2≤0; (x-2)²-(x-2)≤0; (x-2)(x-2-1)≤0; (x-2)(x-3)≤0
x-2=0; x₁=2
x-3=0; x₂=3
4. x²-9≤0
x²≤9 ⇒ x₁=3; x₂=-3
5. (x-1)²+x-7<0
x²-2x+1+x-7<0; x²-x-6<0; x(x-1)<6≤16 ⇒x₁=3; x₂=2
6. x(x-1)≤x+15
x²-x-x≤15 /+1; x²-2x+1≤16; (x-1)²≤16 ⇒
x₁-1=4; x₁=5
x₂-1=-4; x₂=-3
7. (x-1)(x-2)>x-3
x₂-3x+2>x-3; x²-4x+4>-1; (x-2)²>-1
dar,(x-2)²>-1, ∀ x∈R ⇒ (x-2)²>-1
8. (2x-1)²≤9
2x₁-1=3; 2x₁=4; x₁=2;
2x₂-1=-3; 2x₂=-2; x₂=-1
9. (x-2)(x+1)≤3(x+1)
x²-x-2≤3x+3; x²-4x-5≤0 /+9; x²-4x+4≤9; (x-2)²≤9 ⇒
x₁-2=3; x₁=5
x₂-2=-3; x₂=-1
10. (2x-1)(x+1)≤-x+11
2x²-x-1+x-11≤0; 2x²-12≤0 /:2 ⇒ x²≤6
x₁,₂=±[tex]\sqrt{6}[/tex]
11. (x²-1)(x+1)≥0
x²-1≥0; x²≥1 ⇒ x₁,₂=±1
x+1≥0; x)≥-1 ⇒ x₃=-1
Explicație pas cu pas:
