Problem: Terry finds a pile of money with at least 500. If she puts 100 of the pile in her left pocket, gives away $\frac{2}{3}$ of the rest of the pile, and then puts the remaining money from the pile in her right pocket, she’ll have more money than if she instead gave away 500 of the original pile and kept the rest. What are the possible values of the original pile of money?
My solution:
Let x be the pile
$x-100-\frac{2}{3}(x-100)<x-500$
I wrote < since if Terry has more money, the pile is smaller. If Terry has less money, then the pile is bigger. Note: I'm interested in knowing WHY it is wrong.