If $7=x^2+\frac{1}{x^2}$, then what is the greatest possible value of $x+\frac{1}{x}$?
I notice that the given equation involves x^2 and 1/x^2, which are related to x and 1/x by squaring. I wonder if I can use that relationship to rewrite the equation in terms of x and 1/x, and then find their sum. I recall that $(x+\frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2}$, so I can use that to substitute for x^2 and 1/x^2 in the equation. I get $(x+\frac{1}{x})^2 - 2 = 7$, or $(x+\frac{1}{x})^2 = 9$. Now I can take the square root of both sides, but I have to remember that there are two possible values for the square root of 9: 3 and -3. So I have two possible values for x + 1/x: 3 and -3. The question asks for the greatest possible value, so I choose 3 as the answer. 