Part b
Solve the inequality \(|x – 5| \leq 3\) to determine the range of x values that keep the object within safe operational limits.
Inverting a Restricted Absolute Value Function
Given the function \( f(x) = |2x-3| \) with the domain restricted to \( x \geq 1.5 \), find the inverse function \( f^{-1}(y) \) and state its domain.
Proof of the Reverse Triangle Inequality
Prove that for any real numbers \(a\) and \(b\), the following inequality holds: $$ ||a| – |b|| \leq |a – b|. $$ You may use either a direct proof or a proof by contradiction in your solution. Clearly state your approach.
Confirming the Absolute Value Inequality
Prove that for any real number \(x\), the absolute value satisfies \(|x| \geq 0\). In your proof, use the definition of absolute value and consider the cases when \(x \geq 0\) and \(x < 0\).