Question

2 ≤ (x + 1/x − 1) ≤ 3

Answer 1

2 ≤ (x + 1)/(x − 1) ≤ 3     .....x≠1

Para 2 ≤ (x + 1)/(x − 1)

(x + 1)/(x − 1)≥ 2

(x + 1)/(x − 1)-2≥ 0

(x + 1-2x+2)/(x-1)≥ 0

(-x+3)/(x-1)≥ 0

p=-x+3  ..raiz=3  ..a=-1 é decrescente

p+++++++++++++++++++++++(3)------------------------------

q=x-1 ..raiz=1 ..a=1 é crescente

q----------------------------(1)+++++++++++++++++++++++++

Estudo de Sinais:

p+++++++++++(1)---------------------------------------

q----------------------------------(3)+++++++++++++++

p/q----------------(1)+++++++(3)----------------------

(1 , 3]  (i)

Para (x + 1)/(x − 1) ≤ 3

(x + 1)/(x − 1) -3 ≤ 0

(x+1-3x+3)/(x-1) ≤ 0

(-2x+4)/(x-1) ≤ 0

p=-2x+4 ..raiz=2  ..a=-2 é decrescente

p++++++++++++++++++-(2)---------------------------------

q=x-1 ..raiz=1 ..a=1 é crescente

q----------------------------(1)+++++++++++++++++++++++++

Estudo de sinais:

p++++++++++++++++++++(2)---------------------------------

q----------------(1)+++++++++++++++++++++++++

p/q-------------(1)+++++++++(2)-----------------------------

(-∞,1) U [2,+∞)  (ii)

(i) ∩  (ii)

(1 , 3]   ∩ { (-∞,1) U  [2,+∞)}  = [2,3] é a resposta