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 x^2-y^2 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 \left|x^2-y^2\right|+2 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?

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