The Answer Key Says 2 < x < 3

Correct solution: x² − 5x + 6 = (x − 2)(x − 3) > 0. The critical points are 2 and 3. Test points: x = 0 gives 6 > 0 (true); x = 2.5 gives 6.25 − 12.5 + 6 = −0.25 (false); x = 4 gives 16 − 20 + 6 = 2 > 0 (true). Hence the solution is x < 2 or x > 3, equivalently x ∈ (−∞, 2) ∪ (3, ∞). Meera's numerical check is correct and should be praised publicly.

Diagnosis of the error: 2 < x < 3 is the solution of (x − 2)(x − 3) < 0, not > 0. Most students are overgeneralising a remembered pattern ("between the roots") without linking it to the sign of the product. A second route to the same error: students test only x = 2.5, see it "in the middle", and stop. A third, more serious route is an invalid inference from "product is positive": students reason "both factors positive", get x > 3, then also try "both negative" badly and collapse the two cases into a single interval. The error is procedural, not careless - it will recur with rational inequalities and the wavy-curve method unless the sign logic is rebuilt.

Note that the guide is wrong, not merely differently formatted. This matters because students are being trained to trust print over verification.

Principle: Mathematical verification outranks a printed key, and the board examiner marks mathematics, not the guide. But the handling must avoid humiliating students who spent money on the book and avoid a public fight with a publisher before the test.

With students: Do not announce "the guide is rubbish". Put the test-point table on the board and let the class see the contradiction. Then have them paste or write a correction slip on the relevant page: "Correct answer: x < 2 or x > 3. Verified by substitution." Ask students to bring any other answers they doubt to class for checking. Explicitly address the exam belief: "In the board exam, a correct method with correct interval earns marks; a wrong answer copied from a guide loses them."

With parents: A short, factual message works better than a long defence: "In one exercise the guide's answer is incorrect; we have verified the correct answer in class and attached the working. Please ask your child to show you the substitution check." Invite them to verify rather than to trust you.

With other sections: Share a one-page correction through the mathematics teachers, not a circular attacking the publisher. With the principal: Frame it as exam readiness, not controversy: following the guide would cost marks in the chapter test and later. Offer to keep the correction internal to the mathematics department. Do not collect or confiscate the guides; that converts a mathematical issue into a discipline issue.

Activity: sign chart on a number line (15 minutes).

1. (3 min) Factor and mark: Students factor x² − 5x + 6, mark 2 and 3 on a number line drawn on the floor or board, and note that the line is now divided into three regions.
2. (5 min) Test each region: Assign one test value per region (0, 2.5, 4). Students compute the sign of each factor and then of the product. Record +, −, + above the regions. Emphasise that the sign of the product, not the position of the roots, decides the answer.
3. (4 min) Read both directions: Ask for the solution of > 0 and of < 0 from the same chart. This forces students to see that 2 < x < 3 belongs to the negative case. Repeat with a leading coefficient negative, e.g. −(x − 1)(x + 2) > 0, so the rule cannot be memorised as "outside the roots".
4. (3 min) Exit ticket: Solve x² − 7x + 10 < 0 and x² − 4 > 0, and write one sentence explaining why "between the roots" is not a universal rule.

Follow-up: Use Meera's graphical check as a bridge - plot y = x² − 5x + 6 and read where the curve is above the x-axis. Link this to the wavy-curve method for later chapters. Keep the chart displayed for a week and include one interval question in the chapter test so the correction is consolidated, not just discussed.