Equations with |x|: split into sign cases and count roots

CAT has tested this idea 4 times across 3 papers, from 2019 to 2023. Solve one, and the others are the same method with different numbers or wording, which is why repeated concepts are the most efficient thing to practise.

Question	What it asks	Answer
CAT 2023 Slot 1 · QA Q5 near-repeat	2\|x\|(x^2 + 1) = 5x^2; integer solutions	3
CAT 2019 Slot 1 · QA Q29 near-repeat	\|x\|(6x^2 + 1) = 5x^2; number of solutions	5
CAT 2019 Slot 1 · QA Q31	\|x^2 − x − 6\| = x + 2; product of roots	B
CAT 2021 Slot 1 · QA Q8	\|x^2 − 4x − 13\| = r with exactly three roots	C

Part of Inequalities & Modulus (49 questions since 2017).