Can Arithmetic Alone Prove This Inequality?

[NastyAccident]

Homework Statement

(!) Assuming only arithmetic (not the quadratic formula or calculus), prove that
$$\left\{x \in\Re: x^2-2x-3 < 0\right\} = \left\{x \in\Re: -1 < x < 3\right\}$$

Homework Equations

$$\left\{x \in\Re: x^2-2x-3 < 0\right\} = \left\{x \in\Re: -1 < x < 3\right\}$$

The Attempt at a Solution

Again, new and trying to learn how exactly to do this to proper mathematical standards.

I think this will just be a simple plug'n'play:

Test case, 0 and 2 since using boundaries results in 0 < 0:
(0)^2-2(0)-3 < 0
-3 < 0
True

(2)^2-2(2)-3 < 0
4-4-3 < 0
-3 < 0
True

Thoughts? This seemed too easy but when I manipulated with adding and subtracting/multiplying I started to lose focus from the core inequality ie:

x^2-2x-3 < 0
x^2-2x < 3

-1 < x < 3
0 < x+1 < 3+1
0 < x^2+x < 4x

Et cetera...



NastyAccident

 

[ehild]

It is a good start to factorise the expression.

Hint: x^2-2x-3 =(x-1)^2-4.

ehild

 

[gomunkul51]

note: the quadratic formula is achieved through algebra alone. ;)

 

[NastyAccident]

It is a good start to factorise the expression.

Hint: x^2-2x-3 =(x-1)^2-4.

ehild

Going your way:
x^2-2x+1-4 < 0
(x-1)^2-4<0
(x-1)^2<4

-1<x<3
0<x+1<4
x+1<(x-1)^2
x+1<x^2-2x+1
0<x^2-3x
0<x(x-3)
x=0 or x=3?

Somewhere near that?

 

[ehild]

Going your way:
x^2-2x+1-4 < 0
(x-1)^2-4<0
(x-1)^2<4

-1<x<3

Was not that you wanted to arrive at?

It is not that what I meant by factorize, but it is correct up to here.


(x-1)^2-4 = (x-1)^2-2^2=((x-1)-2)((x-1)+2)=(x-3)(x+1)

You have to find the range of x where (x-3)(x+1)<0

The product of two numbers is negative if one of the factors is negative, the other is positive. As x+1>x-3, x-3<0 and x+1>0 that is equivalent to -1<x<3.


ehild

 

[NastyAccident]

Was not that you wanted to arrive at?

It is not that what I meant by factorize, but it is correct up to here.


(x-1)^2-4 = (x-1)^2-2^2=((x-1)-2)((x-1)+2)=(x-3)(x+1)

You have to find the range of x where (x-3)(x+1)<0

The product of two numbers is negative if one of the factors is negative, the other is positive. As x+1>x-3, x-3<0 and x+1>0 that is equivalent to -1<x<3.


ehild

Okay, the way of factoring originally brought up confused me for a second since it appeared to be completing a square to a problem that could of already been factored.

After reading your original suggestion, I had the thought of breaking it down to (x-3)(x+1) >0. Though, after that I was unsure of where to go. And now, with the range comment I am a bit lost as to why you were able to move from (x-3)(x+1)>0 to x+1>x-3. Logically, it does hold since (0)+1>0-3 = 1>-3.

I guess, I'm just getting lost on the transition from (x-3)(x+1)>0 to x+1>x-3 to x-3<0 and x+1>0. In my mind, I see it as going with (x-3)>0 and (x+1) > 0 instead of the inequalities that you presented. If you could clarify that I would appreciate it.



NastyAccident

 

[ehild]

x+1 is always bigger than x-3, isn't it?

I did not say that (x-3)(x+1)>0. You have to find the values of x which make (x-3)(x+1)<0 true, that is the product of (x+1) and (x-3) is negative. That happens if one of the factors is negative, the other positive. As x+1>x-3, if x+1<0, x-3 is also negative and their product is positive. So x-3 has to be negative and x+1 positive:

x-3<0 --->x<3

and x+1>1 --->x>1


ehild

 

[NastyAccident]

x+1 is always bigger than x-3, isn't it?

I did not say that (x-3)(x+1)>0. You have to find the values of x which make (x-3)(x+1)<0 true, that is the product of (x+1) and (x-3) is negative. That happens if one of the factors is negative, the other positive. As x+1>x-3, if x+1<0, x-3 is also negative and their product is positive. So x-3 has to be negative and x+1 positive:

x-3<0 --->x<3

and x+1>1 --->x>1


ehild

First, I apologize for the (x-3)(x+1)>0 statement. It should have been (x-3)(x+1) < 0. I hit the wrong key.

I now see and understand why the split is occurring like that since you are trying to find terms that create a negative which would yield something that is less than zero.

Now, this proves that set A (x^2-2x-3 < 0) is included in Set B (-1<x<3). In order to ensure that both are equal sets, I now have to repeat this process so that set B is included in Set A.

So, would it be fair to do the following:

-1<x<3
-1<x
x<3
0<x+1
x-3<0
(x+1)(x-3)<0
x^2-2x-3<0

Hence,
Set A = Set B



NastyAccident

 

[ehild]

Well done! ehild