Question

решите неравенство 5(х-6√(х)+8)/(х-16)<=√(х) -2

Answer 1

5*(x-6√x+8)/(x-16)≤√x-2     ОДЗ:  x≥=0   x-16≠0  x≠16   x∈[0;16)U(16;+∞)
Пусть √x=t≥0   ⇒
5*(t²-6t+8)/(t²-16)≤t-2
t²-6t+8=0   D=4   t₁=2   t₂=4
5*(t-2)(t-4)/((t-4)(t+4))≤t-2
5*(t-2)/(t+4)-(t-2)≤0
(t-2)(5/(t+4)-1)/≤0
(t-2)(5-t-4)/(t+4)≤0
(t-2)(2-t)/(t+4)≤0
(√x-2)(1-√x)/(√x+4)≤0
√x-2=0  √x=2  x=4
1-√x=0  √x=1  x=1
√x+4>0  ⇒
-∞___-____[0]___-___1_______+______4____-_____(16)_____-____+∞
x∈[0;1]U[4;16)U(16;+∞).