Răspuns:
a) x ∈ (-∞; 1/2] ∪ [3; +∞)
b) x ∈ (-3/2; -1] ∪ [2; +∞)
c) x ∈ (-∞; 5/2] ∪ (3; +∞)
d) x ∈ (2/3; 2)
Explicație pas cu pas:
a) 1 - 2x = 0 ⇒ 2x = 1 ⇒ x = 1/2
x - 3 = 0 ⇒ x = 3
Tabel de semn:
-∞................1/2....3.......+∞
(1-2x)+++++0-------------
(x-3)----------------0++++
f(x)-----------0+++0------
⇒ x ∈ (-∞; 1/2] ∪ [3; +∞)
b) x + 1 = 0 ⇒ x = -1
x - 2 = 0 ⇒ x = 2
2x + 3 = 0 ⇒ 2x = -3 ⇒ x = -3/2
Tabel de semn:
-∞................-3/2....-1.......2......+∞
(x+1)------------------0++++++++
(x-2)--------------------------0+++
(x+1)(x-2)+++++++0------0+++
(2x+3)---------0++++++++++++
f(x)--------------|+++0------0+++
⇒ x ∈ (-3/2; -1] ∪ [2; +∞)
[tex]c) \ \dfrac{x-2}{x-3} + 1 \geq 0 \Rightarrow \dfrac{x-2+x-3}{x-3} \geq 0 \Rightarrow \dfrac{2x-5}{x-3} \geq 0[/tex]
2x - 5 = 0 ⇒ 2x = 5 ⇒ x = 5/2
x - 3 = 0 ⇒ x = 3
Tabel de semn:
-∞................5/2.......3........+∞
(2x-5)---------0++++++++++
(x-3)--------------------0++++
f(x)++++++++0-------|+++++
⇒ x ∈ (-∞; 5/2] ∪ (3; +∞)
[tex]d) \ \dfrac{2x}{x-2} - \dfrac{1}{2} < 0 \Rightarrow \dfrac{4x-x+2}{2(x-2)} < 0 \Rightarrow \dfrac{3x+2}{2(x-2)} < 0[/tex]
3x + 2 = 0 ⇒ 3x = -2 ⇒ x = -2/3
x - 2 = 0 ⇒ x = 2
Tabel de semn:
-∞................-2/3.......2........+∞
(3x+2)---------0++++++++++
(x-2)--------------------0++++
f(x)++++++++0-------|+++++
⇒ x ∈ (2/3; 2)
