Question

Inequação Modular

| x + 2 / 2x - 1 | >= 1

Answer 1

Resposta:

Se for | (x+2)/(2x-1)| ≥ 1

Caso (x+2)/(2x-1) ≥ 0

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

-------------------------(1/2)++++++++

++++(-2)-------------(1/2)++++++++

-2 ≥ x > (1/2)

Podemos tirar o sinal de módulo

(x+2)/(2x-1) ≥ 1

(x+2)/(2x-1) -1 ≥ 0

(x+2-2x+1)/(2x-1) ≥ 0

(-x+3)/(2x-1)≥ 0

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

----------(1/2)+++++++++++++++++

----------(1/2)+++++(3)--------------------

1/2 < x  ≤ 3   (i)

Caso (x+2)/(2x-1) < 0

-2 ≤ x < (1/2)

-(x+2)/(2x-1) ≤1

(x+2)/(2x-1)≤-1

(x+2)/(2x-1) +1 ≤0

(x+2+2x-1)/(2x-1) ≤0

(3x+1)/(2x-1) ≤ 0

-----------------(-1/3)++++++++++++++

-------------------------------(1/2)++++++

++++++++++(-1/3)--------(1/2)++++++

-1/3 ≤ x < (1/2)

-1/3 ≤ x <1/2   (ii)

Resposta:  1/2 < x  ≤ 3  U  -1/3 ≤ x <1/2  = [ -1/3 ; 1/2) U (1/2 ; 3]