2024 HSC Mathematics Extension 1
Question 12(e):
Absolute value inequality
The diagram shows the graph of (vertical asymptote , positive branch on each side).
For what values of is ?
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
- Note (denominator). Also always where defined, so the inequality forces hence (actually still requires right-hand side positive so ; at LHS fails).
- Case : , so . Both sides positive for : .
- For , this holds when . So solution piece .
- Case : , so . For this with positive RHS need , so : .
- Hence (which lies inside ).
- Combined solution: .
- NESA 12(e): 3 marks correct; 2 marks for critical points ; 1 mark for recognising two cases, or that is critical, or graphing both curves.
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
For with , , find all solutions of the equality.