Determine all real number x that satisfy |x^2-x-3|>|x^2-x-3|?

Answer 1

Any easy way to deal with absolute value inequalities is to treat them like equations to begin with.

You have 2 parts:

x^2 -x -3 = x^2 -x -3
They are identical, so everything cancels here.

Now take the opposite of the second polynomial.

x^2 -x -3 = -x^2 +x + 3
2x^2 -2x -6 =0
factor out the common 2

x^2 -x - 3 = 0
now solve for x
this will give you the boundaries for the solution. Since this is a greater than inequality... look for your answers to be off to the right and left of your values.

For less than, the answer would fall in between.