HELP - solving an inequality

[shulmanovsky]

Hi,
I'm not sure i'm posting in the right section...
Anyway, i have an inequality i'm trying to solve forever:

[MATH]\left \lfloor \left | x+1 \right |-\left | x \right | \right \rfloor\geq x^2[/MATH]
I divided it into 4 options:
1. [MATH](x\geq 0)\wedge (x\geq -1)[/MATH]which gives me x is between 0 and 1.
2. [MATH](x\geq 0)\wedge (x< -1)[/MATH]which isn't possible
3. [MATH](x< -1)\wedge (x< 0)[/MATH]which gives me [MATH]x^2\leq -1[/MATH] and it's impossible too
but when i check for [MATH]-1\leq x< 0[/MATH] i get [MATH]\left \lfloor 2x+1 \right \rfloor>x^2[/MATH]i dont know how to continue.
thank you for your help!

 

[pka]

Have you looked at the plot? The look at this one. Then look here

 

[LCKurtz]

Here's another picture:

 

[Steven G]

In terms of your 4 cases you got I think you did not do it efficiently.

x + 1 = 0 when x=-1 and well x=0 when x=0.

This divides the x-axis into three regions: x<-1, -1<x<0 and x>0

 

[shulmanovsky]

In terms of your 4 cases you got I think you did not do it efficiently.

x + 1 = 0 when x=-1 and well x=0 when x=0.

This divides the x-axis into three regions: x<-1, -1<x<0 and x>0

Yes, it does divide in to three cases, one of the four cases is just impossible so it doesn't count. I'm stuck at the case when x is between 0 and -1. I just don't know what to do with the floor function.

 

[Steven G]

Forget about the floor function for a moment. What does |x+1| - |x| equal when -1<x<0?

What does |x+1| equal if -1<x<0 (hint: either it is x+1 or -(x+1))
What does |x| equal if -1<x<0 (hint: it is either x or -x).

Now add the two terms. What do you get? What is the floor of that function?

 

[shulmanovsky]

Forget about the floor function for a moment. What does |x+1| - |x| equal when -1<x<0?

What does |x+1| equal if -1<x<0 (hint: either it is x+1 or -(x+1))
What does |x| equal if -1<x<0 (hint: it is either x or -x).

Now add the two terms. What do you get? What is the floor of that function?

When x is between 0 and -1, |x+1|-|x| equals 2x+1. I'm stuck at the point where I add the floor function. floor(2x+1)≥x^2

 

[Steven G]

When x is between 0 and -1, |x+1|-|x| equals 2x+1. I'm stuck at the point where I add the floor function. floor(2x+1)≥x^2

OK, you have 2x+1 when -1<x<0.
-1<x<0
-2<2x<0
-1<2x+1<1
Now find the restriction on x such that -1<2x+1<0 (what is the floor of that?) and 0<2x+1<1 (what is the floor of that?)