2. You and your math friends, Andy and Betty, are asked to consider the inequality x^2 ≥ x where x is a real number.
2.a) Your friend Andy says: Yes, this inequality is always true! Look, let's try a few number.. we can see that...
1^2 = 1 ≥ 1
2^2 = 4 ≥ 2
3^2 = 9 ≥ 3
see? It will work for any real number.
Describe as many mistakes as you can in Andy's argument.
2.b) Your friend Betty comments on Andy's work: No Andy, this inequality isn't true, you goose, you forgot about negative numbers! Consider what happens when we look at -2. We can see that...
-2^2 = -4
and -4 < 2, so it isn't always true!
Betty's conclusion is correct, but her argument is wrong. Describe the mistake in Betty's argument.
2.c) Give Andy and Betty the cold hard truth, give them a legitimate example that shows x^2 ≥ x is not true for all real numbers x (i.e., give a counter example).
2.d) Give the solution set to x^2 ≥ x in interval notation. In other words, describe the set of real numbers x such that x^2 ≥ x is true.
2.e) In interval notation, describe the set of x values such that x^2 ≥ x is not true. In other words, describe the set of real numbers x such that x^2 < x.