M26-25

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

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.