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Решение смотри в приложениях
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в
x-2/x=a
a²-2a-3≥0
a1+a2=2 U a1*a2=-3
a1=-1 U a2=3
a≤-1⇒x-2/x≤-1
(x²+x-2)/x≤0
x1=x2=-1 U x1*x2=-2⇒x1=-2 U x2=1
x=0
_ + _ +
-------------[-2]------------(0)-----------------[1]----------------------
x∈(-∞;-2] U (0;1]
a≥3⇒x-2/x≥3
(x²-3x-2)/x≥0
D=9+8=17
x1=(3-√17)/2 U x2=(3+√17)/2
x=0
_ + _ +
-------------[(3-√17)/2]------------(0)-----------------[(3+√17)/2]---------------------
x∈[(3-√17)/2;0) U [(3+√17)/2;∞)
Ответ x∈(-∞;-2] U [(3-√17)/2;0) U (0;1] U [(3+√17)/2;∞)
г
x/(x²-2)=a
a+6/a>7
(a²-7a+6)/a>0
a1+a2=7 U a1*a2=6⇒a1=1 U a2=6
a=0
_ + _ +
---------------------(0)-------------------(1)------------------(6)-------------------
0<a<1 U a>6
{x/(x²-2)>0
{x/(x²-2)<1
1)x/(x-√2)(x+√2)>0
x=0 x=√2 x=-√2
_ + _ +
-----------------(-√2)-------------------(0)------------------(√2)-----------
-√2<x<0 U x>√2
2)x/(x-√2)(x+√2)-1<0
(x-x²+2)/(x-√2)(x+√2)<0
(x²-x-2)/(x-√2)(x+√2)>0
x1+x2=1 U x1*x2=-2⇒x1=-1 U x2=2
x=√2 x=-√2
x<-√2 U -1<x<√2 U x>2
x∈(-1;0) U (2;∞)
x/(x-√2)(x+√2)>6
(x-6x²+12)/(x-√2)(x+√2)>0
(6x²-x-12)/(x-√2)(x+√2)<0
D=1+288=289
x1=(1-17)/12=-4/3 U x2=(1+17)/12=3/2
x=-√2 x=√2
+ _ + _ +
-----------------(-√2)--------(-4/3)-----------(√2)------------------(3/2)-----------
-√2<x<-4/3 U √2<x<3/2
Ответ x∈(-√2;-4/3) U (-1;0) U (√2;3/2) U (2;∞)
x-2/x=a
a²-2a-3≥0
a1+a2=2 U a1*a2=-3
a1=-1 U a2=3
a≤-1⇒x-2/x≤-1
(x²+x-2)/x≤0
x1=x2=-1 U x1*x2=-2⇒x1=-2 U x2=1
x=0
_ + _ +
-------------[-2]------------(0)-----------------[1]----------------------
x∈(-∞;-2] U (0;1]
a≥3⇒x-2/x≥3
(x²-3x-2)/x≥0
D=9+8=17
x1=(3-√17)/2 U x2=(3+√17)/2
x=0
_ + _ +
-------------[(3-√17)/2]------------(0)-----------------[(3+√17)/2]---------------------
x∈[(3-√17)/2;0) U [(3+√17)/2;∞)
Ответ x∈(-∞;-2] U [(3-√17)/2;0) U (0;1] U [(3+√17)/2;∞)
г
x/(x²-2)=a
a+6/a>7
(a²-7a+6)/a>0
a1+a2=7 U a1*a2=6⇒a1=1 U a2=6
a=0
_ + _ +
---------------------(0)-------------------(1)------------------(6)-------------------
0<a<1 U a>6
{x/(x²-2)>0
{x/(x²-2)<1
1)x/(x-√2)(x+√2)>0
x=0 x=√2 x=-√2
_ + _ +
-----------------(-√2)-------------------(0)------------------(√2)-----------
-√2<x<0 U x>√2
2)x/(x-√2)(x+√2)-1<0
(x-x²+2)/(x-√2)(x+√2)<0
(x²-x-2)/(x-√2)(x+√2)>0
x1+x2=1 U x1*x2=-2⇒x1=-1 U x2=2
x=√2 x=-√2
x<-√2 U -1<x<√2 U x>2
x∈(-1;0) U (2;∞)
x/(x-√2)(x+√2)>6
(x-6x²+12)/(x-√2)(x+√2)>0
(6x²-x-12)/(x-√2)(x+√2)<0
D=1+288=289
x1=(1-17)/12=-4/3 U x2=(1+17)/12=3/2
x=-√2 x=√2
+ _ + _ +
-----------------(-√2)--------(-4/3)-----------(√2)------------------(3/2)-----------
-√2<x<-4/3 U √2<x<3/2
Ответ x∈(-√2;-4/3) U (-1;0) U (√2;3/2) U (2;∞)
Приложения:
![](https://st.uroker.com/files/6e3/6e3b6b781fc9681adf458fe15dc03809.jpg)
![](https://st.uroker.com/files/166/16600c9471fd2cb1f5bf8a8f13d0cb47.jpg)
![](https://st.uroker.com/files/239/23939b3f0259b15d1e7054dc24675948.jpg)
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