I am very confused on how to find the absolute value of an equation without graphing it first. Can you help me?

The definition of absolute value is |x| = { x if x >=0; or -x if x < 0 }.

Note that sqrt(x^2) = |x|.

E.g., if y = x^2 - 1, find |y|.

Case 1: If x^2 - 1 < 0, or x^2 < 1, or sqrt(x^2) < sqrt(1), or |x| < 1, or -1 < x < 1,

then |y| = -y = 1 - x^2.

Case 2: If x^2 - 1 >= 0, or x <= -1 or x >= 1,

then |y| = +y = x^2 - 1.

|y| is a piecewise function: |y| = { 1 - x^2 if -1 < x < 1; or x^2 -1 otherwise }.

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