I’m just going to make up a symbol for better visualization. The symbol will look like this: #
For all numbers x and y, let x # y be defined by x # y= |x^2-y^2| + 2. What is the smallest possible value of x # y?
This was a 2/5 on the difficulty scale yet I somehow didn’t understand this and still got it wrong. I tried to do some weird algebra that got me nowhere so I moved on. Funny thing this was the only question I got wrong in the section.
Think about the least possible value you can have inside absolute value brackets, using a simpler expression. What’s the least possible value of |x|? When x = 0, then |x| = 0. For any other value of x, |x| will be positive, so the least possible value of |x| is 0.
So far, so good? If the least possible value of |x| is 0, then what’s the least possible value of |x| + 2? It’s 2, right? It’s gotta be.
The same thing is going on here. The question says for all values of x and y, so that means x and y can be equal, which would make equal 0. So it’s possible to have a 0 in the absolute value brackets, and that’s by definition the least value you can have in any absolute value brackets. From there, the least possible value of the whole expression is 2.
So anytime I see a question that says for all values of x and y, or at least wording that implies that case, I can safely assume both values at 1 point or another can equal each other?
Yes, unless the question tells you otherwise (e.g. it says “x≠y”).