2022 HSC Mathematics Extension 1
Question 11(f):
Absolute value inequality
Solve .
How to recognise this question
Question type: absolute value inequalities
- The command and notation point to absolute value inequalities.
- The requested response is a worked solution.
How to handle it: separate the absolute-value statement into its valid algebraic cases.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Domain: (denominator zero).
- Bring to one side: .
- Equivalently multiply by the always-nonnegative square (for ): , then after careful rearrangement as in the NESA sample, yielding .
- Critical points and . Sign chart: the product on , but is excluded.
- Solution: , i.e. .
- NESA: 3 marks correct; 2 for correct critical values; 1 for excluding or attempting to handle the denominator.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
Solve .
Question 2
Solve .
Question 3
Solve by multiplying through by .