One thing that I have difficulty with is how to know how to manipulate the equations or inequalities within the proofs. For example, look at Result 5.18:
Result 5.18 Let x be a nonzero real number. If x + (1/x) < 2, then x < 0.
Direct Proof Assume that x + (1/x) < 2. Since x does not equal 0, we know that x^2 > 0. Multiplying both sides of the inequality x + (1/x) < 2 by x^2, we obtain x^2(x + (1/x)) < 2x^2. Simplifying this inequality, we have x^3 + x − 2x^2 < 0; so x(x^2− 2x + 1 ) = x(x − 1)^2 < 0.
The proof concludes shortly thereafter. My problem is not understanding that the proof is valid, but, rather, it is how would I know to multiply both sides of the inequality by x^2? Is this simply a trick-turned-technique? Should I memorize this method? Maybe multiplying by x would make sense to me, considering you have a (1/x), but the multiplying by x^2 just stood out to me as unique. "That's great and all that it worked out," I thought to myself. "But how would I have come up with that idea to multiply by x^2 on my own?"
The "story" problem with the three prisoners was brilliant. I really liked it. My 6-year old son entered the room right when I was re-reading and pondering the problem and asked, "What are you studying?" I told him there was a story problem about prisoners and he wanted to hear it. After hearing the story, my son answered the question about the prisoner correctly identifying the color of his dot: "Maybe he just saw the other two guys with red dots and thought, 'I probably have a red dot, too.'" Sounded like a good explanation to me! It was satisfying to me that, only once I was explaining it out to my son, the answer to the question correctly came to my mind. I had figured it out - and only had done so once I started to put into words what I was thinking. This illustrated something to me that I didn't realize: talking a problem out can be a powerful way to help you arrive at a desired answer or result.
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