epravetz
I'm having a really hard time understanding why it's y-1 and not -1 - y
I get to the point of having -1 - √(y^2)
I don't understand how |y| = -y (can ever equal that), because the square root of y^2 is y, and if y is assumed negative then shouldn't you just leave it as y, why would you have to add a negative in front of it. Take y=-2, then -y=2 and subsequently not be negative? I'm just stuck on this logic
Firstly, \(\sqrt{y^2}=|y|\). This is because the square root function always returns non-negative values, and |y| ensures that the result is non-negative for all y.
For instance, consider y = -5:
\(\sqrt{(-5)^2} = \sqrt{25} = 5\), which is |-5|.
Next, when \(y \geq 0\), |y| is simply y. This is because y is already non-negative, so there's no need for adjustment.
For instance, if \(y = 3\):
\(|3| = 3\)
However, when \(y < 0\), |y| is -y to make it non-negative.
For example, for \(y = -2\):
\(|-2| = -(-2) = 2\)
Thus, this property ensures that the result of |y| is always non-negative, regardless of the original sign of y.
The above are basic properties of absolute values, so I suggest brushing up on fundamentals before solving hard questions on modulus:
10. Absolute Value
For more check Ultimate GMAT Quantitative Megathread
Hope it helps.