Question

I'm having trouble with inequalities. Basically my problem is about which way the sign goes. Here's an example..

X(X+1)>2

so then it becomes

X^2 + X>2

Correct? Then

X^2 + X - 2>0

then factor so it becomes

(X+2)(X-1)>0

And so I end up with X>-2 and X>1 as my answer, but the book says it should be X<-2. Why did the sign switch? I didn't divide by a negative did I? Can someone help me please?

Answer 1

Here is the logical analysis:

(X+2)(X-1)>0 says that the answer is positive, so either

x+2 > 0 AND x-1 > 0
x > -2 AND x > 1 ----> x > 1
<b>or</b>
x+2 < 0 AND x-1 < 0
x < -2 AND x < 1 ----x < -2

so x < -2 OR x > 1

I use the following quick way to solve these kind of problems.

from (X+2)(X-1)>0
the "critical values", or the solution to the corresponding "equation", are
x = -2 and x = 1

put these values on a number line, this will split the number line into 3 regions
1. x < -2
2. x between -2 and 1
3. x > 1

select any value of x in those regions and test it in your factored inequation.

e.g. for
1. I picked x=-5 that works
2. let x = 0 , does NOT work
3. let x = 10 , that also works

so x > 1 OR x < -2