g10-03-03 · Core topics
Do the endpoints belong?
What is the difference between < and ≤ at a root?
1 MIN · PREDICT ON PAPER
Keep the same polynomial and change only the inequality symbol.
Discuss or write first. No typed answer is required to continue.
Solution: (1, 4)
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
A root gives exactly zero, so it belongs precisely when equality is allowed. Repeated roots need their own sign analysis. Finite displayed cases suggest a conjecture; the justification below states why it holds under the stated conditions.
Conditions: Use a(x−r)(x−s) with real parameters, including a=0.
Mathematical justification: At any nonroot the strict and non-strict versions have the same sign test. At each root the value is 0; comparing 0 with 0 gives its inclusion exactly.
1 MIN · EXIT CHECK
Solve (x−2)²<0 and (x−2)²≤0.
Check after answering
The first has no solution; the second has the singleton {2}.
Teacher guidance and extension
Before: Root signs and endpoint notation
Ask: Ask pupils to substitute each endpoint instead of memorising bracket patterns.
Watch for: Open endpoints are not approximate exclusions; they exclude exactly the root.
Challenge: Classify all four inequalities for (x−2)².
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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