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.

A 0

B 1

C 2

D 3

E 4

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.

## Comments (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”).