For no solutions: .
For one solution: .
For two solutions: .
Let's start with where
So for one solution, or .
It's a little harder working out the values of for which there are no solutions or two solutions.
Have you heard of the absolute value function?
if and if .
In other words, represents the SIZE of (if you were to draw a number line and measure the distance from to ).
Now, since , that means that is something that has SIZE .
This means .
This is particularly useful if dealing with squares in inequalities.
Now back to the problem.
Let's find the value of for which there are not any solutions.
Now to undo this square, we need to take the square root:
But , so
So, this means that the "size" of has to be less than 1.
This means . You can verify this by drawing a number line (the distance of any of the points in this region from 0 is less than 1).
So for there to be no solutions, .
Finally, for 2 solutions:
So the size of has to be greater than 1.
This means or .
Once again, you can check this with a number line. The distance from any point or to is obviously going to be .
Therefore, for 2 solutions: or .
0 solutions: , which is the same as .
1 solution: , which is the same as or .
2 solutions: , which is the same as or .