Question

4x-x²小于5|2x+3|-1小于等于0？

Answer 1

4x-x^2<5|2x+3|-1<=0,
各加1，得1+4x-x^2<5|2x+3|<=1,(1)
1+4x-x^2=5-(x-2)^2,
当x>2+√5或x<2-√5.....................(2)
时1+4x-x^2<0,(1)化为
-1<=5(2x+3)<=1,
-1.6<=x<=-1.4满足(2).
当2-√5<=x<=2+√5...................(3)
时1+4x-x^2>=0,(1)化为
1+4x-x^2<10x+15<=1或-1<=10x+15<x^2-4x-1,
分别化为x^2+6x+14>0,10x<=-14;.......................(4)
-16<=10x,x^2-14x-16>0........................................(5)
(4)的解:x<=-1.4,不满足(3),
由(5),x>=-1.6,x>7+√65或x<7-√65，
求交集得 x>7+√65，不满足（3）。
综上，-1.6<=x<=-1.4.

Answer 2

4x-x^2<|2x+3|-10≤0
|2x+3|-10≤0 (1)
4x-x^2<|2x+3|-10 (2)
from (1)
|2x+3|-10≤0
|2x+3|≤10
-10≤2x+3≤10
-13≤2x≤7
-13/2≤x≤7/2 (3)
from (2)
4x-x^2<|2x+3|-10
|2x+3|>10+4x-x^2
case 1: x≥-3/2
|2x+3|>10+4x-x^2
2x+3>10+4x-x^2
x^2-2x-7>0
x>1+2√2 or x<1-2√2
solution for case 1: x>1+2√2
case 2: x<-3/2
|2x+3|>10+4x-x^2

-(2x+3)>10+4x-x^2
x^2-6x-13>0
x>3+√22 or x<3-√22
solution for case 2: x<3-√22
ie
4x-x^2<|2x+3|-10
x<3-√22 or x>1+2√2 (4)
//
4x-x^2<|2x+3|-10≤0
(3) and (4)
-13/2≤x≤7/2 and "x<3-√22 or x>1+2√2"
-13/2≤x<3-√22 or 1+2√2<x≤7/2