Question

不等式log2 (x+1/x+6)≤3的解集为＿＿＿＿

答案是｛x|-3-2√2＜x<-3+2√2或x=1}

Answer 1

┏x+1/x+6>0 ①这是为了保证对数有意义
┗x+1/x+6≤2^3 ②
①<=>(x^2+6*x+1)/x>0
<=> [x-(-3-2√2)]*[x-(-3+2√2)]/x>0
<=> -3-2√2 <x<-3+2√2 或x>0

②<=>(x^2-2*x+1)/x≤0
<=> (x-1)^2/x≤0
<=> x=1 或x<0

因此,原不等式的解集为:
(-3-2√2,-3+2√2)∪{1}

Answer 2

原式等价于0<X+1/X+6≤2^3,分别假设X<0,X>0,不等式两边乘X，化为整式，就是2个一元二次不等式组，就简单了。

Answer 3

-47/7<=X<=-6