Simplify the expression for the given values of \( x \). \( |x-1|-|x-2| \) if \( x \geq 4 \) Answer:

Given the expression \[ |x-1| - |x-2| \] and knowing that \( x \geq 4 \), we analyze each absolute value term. 1. For \( x \geq 4 \), the expression \( x-1 \) is non-negative because \( x-1 \geq 3 \). Thus, we have: \[ |x-1| = x-1. \] 2. Similarly, for \( x \geq 4 \), the expression \( x-2 \) is also non-negative because \( x-2 \geq 2 \). Thus, we have: \[ |x-2| = x-2. \] Substitute these back into the original expression: \[ |x-1| - |x-2| = (x-1) - (x-2). \] Simplify the expression: \[ (x-1) - (x-2) = x - 1 - x + 2. \] Combine like terms: \[ x - x - 1 + 2 = 1. \] Thus, for \( x \geq 4 \), \[ |x-1| - |x-2| = 1. \]