Question

Resolva em R as inequações:
a) 1/(x-4) < 2/(x+3)

b) (x+1)/(x+2) > (x+3)/(x+4)

Answer 1

a)

1/(x-4) < 2/(x+3)

1/(x-4) -2/(x+3) < 0

(x+3-2x+8)/(x-4)*(x+3)  <0

(-x+11)/(x²+3x-4x-12) < 0

(-x+11)/(x²-x-12) < 0

p=-x+11 ==>raiz x=11    e a=-1<0 decrescente

p+++++++++++++(11)--------------------------------------

q=x²-x-12  raízes x'=4 e x''=-3  ...a=1>0 ...conc/cima

q++++++++++++(-3)----------------------(4)+++++++++++++++++

Estudo de sinais:

p+++++++++++++++++++++++++++++(11)--------------

q++++++++(-3)-----------(4)+++++++++++++++++

p/q++++++(-3)-----------(4)+++++++++(11)--------------

-3 < x <4  U x>11

b)

(x+1)/(x+2) > (x+3)/(x+4)

(x+1)/(x+2) - (x+3)/(x+4) >0

(x²+4x+x+4-x²-3x-2x-6)/(x+2)*(x+4) >0

(-2)/(x²+4x+2x+8) >0

2/(x²+6x+8) < 0

p=x²+6x=8 ..raízes==>x'=-4 e x''=-2 ..a=1>0 ..conv/cima

p++++++++++++(-4)------------------------(-2)+++++++++++++

-4 < x < -2   ou (-4 , -2)