Question

x-1)(x-2)<1-x
Răspunsul trebuie sa fie (-2,-1)​

Answer 1

Răspuns:

Explicație pas cu pas:

(x-1)(x-2) < 1-x <=>

x²-2x-x+2 < 1-x <=>

x²-3x + 2 - 1 + x < 0  =>

x²-2x + 1 < 0  <=>

(x-1)² < 0  

(x-1)² ≥ 0 ; (∀) x ∈ R =>

Solutie : x = ∅

(x-1)(x+2) < 1-x <=>

x²+2x-x-2 < 1-x <=>

x²+x+x -2-1 < 0 <=>

x²+2x-3 < 0

x²+2x-3 = 0  ;   a = 1 ; b = 2 ; c = -3

Δ = b²-4ac = 2²-4·1·(-3) = 4+12 = 16

√Δ = √16 = 4

x₁,₂ = (-b±√Δ)/2a = (-2±4)/2

x₁ = (-2-4)/2 = -6/2 = -3

x₂ = (-2+4)/2 = 2/2 = 1

          x I -∞               -3         1             +∞

x²+2x-3 I ++++++++++0--------0++++++++ =>

x ∈ (-3 ; 1)

Answer 2

$\it (x-1)(x+2)<1-x \Rightarrow (x-1)(x+2)+x-1<0 \Rightarrow (x-1)(x+2+1)<0 \Rightarrow \\ \\ \Rightarrow (x-1)(x+3)<0 \Rightarrow x\in(-3,\ 1)$