-1<=[x]-x^2<=2 Solve the in equality..... [.] denotes greater integer function.

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-1<=[x]-x^2<=2 Solve the in equality..... [.] denotes greater integer function.

[From: ] [author: ] [Date: 12-07-05] [Hit: ]

Let x∈(n,n+1] where n=3,4,5,........

-1 ≤ [x]-x² ≤ 2

Let x=0
[0]-0²=0 and -1 ≤ 0 ≤ 2 => x∈{0}

Let x∈(0,1]
[x]=1 and 0 Hence
[x]-x²=1-x² and -1 ≤ 1-x² ≤ 2 => x∈(0,1]

Let x∈(1,2]
[x]=2 and 1 Hence
-1 < 2-x² ≤ 4
-3 < -x² ≤ 2 => -3 < -x² => x² < 3 => x∈(1,√3]

Let x∈(2,3]
[x]=3 and 4 Hence
-1 < 3-x² ≤ 4
-4 < -x² ≤ 1 => x² < 4 => x∉(2,3]

Let x∈(n,n+1] where n=3,4,5,...
[x]=n+1 and n² Hence
-1 < n+1-x² ≤ 4
-2-n < -x² ≤ 3 - n
-2-n < -x² => x² < 2+n but n² < x²
Hence
n² < 2+n => n²-n-1 < 0 => n≠3,4,5,..

Let x∈(-1,0)
[x]=0 and 0 -1 ≤ -x² ≤ 2 => -1 ≤ -x² => x² ≤ 1 => x∈(-1,0)

Let x∈(-2,-1]
[x]=-1 and 1≤x²<4
-1 ≤ -1-x² ≤ 2
0 ≤ -x² ≤ 1
Hence
-x² ≤ 1 => x² ≥ -1
and
0 ≤ -x² => x∉(-2,-1]

Let x∈(n,n+1] where n=-2,-3,-4,...
[x]=n+1 and (n+1)²≤x² Hence
-1 < n+1-x² ≤ 4
-2-n < -x² ≤ 3-n
-2-n < -x² => x² < 2+n but (n+1)²≤x²
Hence
(n+1)²≤x²< 2+n
(n+1)²< 2+n
n²+2n+1< 2+n
n²+n-1< 0 => n≠-2,-3,-4,...

Answer
x∈(-1,√3]

1

keywords: equality,Solve,integer,greater,function,denotes,in,lt,the,-1<=[x]-x^2<=2 Solve the in equality..... [.] denotes greater integer function.

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