Absolute value inequality — Strict outside roots

Question

Solve .

This practice question was inspired by Question 11(b) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: separate the absolute-value statement into its valid algebraic cases.Factor .
Finish the calculation, then check that the result meets the question’s conditions.Positive outside the roots, endpoints excluded.
State the final answer clearly in the form the question requested.Answer: or .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: At the product is negative, so interior is not in the solution.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.