Chapter 1.4, Problem 9E

Let a, b, and c be integers and x, y, and z be real numbers. Use the technique of working backward from the desired conclusion to prove that
(a) if x and y are nonnegative, then $\frac{x+y}{2}\ge \sqrt{xy}$ .
Where in the proof do we use the fact that x and y are nonnegative?
(b) if a divides b and a divides $b+c$ , then a divides 3c.
(c) if $ab>0$ and $bc<0$ , then $a{x}^{\text{2}}+bx+c=0$ has two real solutions.
(d) if $x^{\text{3}}+\text{2}{x}^{\text{2}}<0$ , then $\text{2}x+\text{5}<\text{11}$ .
(e) if an isosceles triangle has sides of length x, y, and z, where $x=y$ and $z=\sqrt{\text{2}xy}$ , then it is a right triangle.
(f) if x and y are positive and $x^{\text{2}}{y}^{\text{2}}-\text{25}{x}^{\text{2}}-\text{9}{y}^{\text{2}}+\text{225}<0$ , then $x<\text{3}$ and $y>\text{5}$ or else $x>\text{3}$ and $y<\text{5}$ .