Non linear inequalities in one variable?

Question

can someone explain this in more detail i got confused all 3 examples here they are http://www.flickr.com/photos/26479899@N03/2783900305/

Answer 1

For the first two parts remember alpha < beta is given above.

1. For the product to be positive you need either both brackets to be positive or both negative. If x < alpha both are negative. If x > beta both are positive.

2. For the product to be negative one bracket must be negative but not the other. If x is between alpha and beta this must be true because (x - beta) will be negative but (x - alpha) will be positive.

3. A real number squared must be positive or zero. If the result is zero then the number you are squaring must be zero. This can only happen if x = alpha.