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LESSON 7 OF 7 · FREE · ABOUT 5 MINUTES

Use cases to test what must be true

Distinguish a proof from an example and preserve sign, boundary and counting restrictions.

The idea

A quantitative comparison asks which relationship holds for every permitted situation. One numerical example can suggest a relationship, but it cannot usually prove that the same relationship always holds. To show that the relationship is undetermined, find two valid cases producing different comparison outcomes. Equality in one case and a strict inequality in another is already enough; you do not need to produce all three outcomes. Before choosing values, write the constraints. A real variable may be negative or fractional unless stated otherwise; an integer variable may not. A positive length cannot be zero. Strict inequalities exclude equality boundaries. Examples that violate even one condition tell you nothing about the required comparison. Convenient numbers are useful only after these restrictions are respected. When a relationship appears fixed, look for a short proof. Subtract the two quantities and study the sign of the difference. Factor expressions to expose signs, or use geometry to turn a length relation into an inequality. For counting, replace trial and error with an exhaustive case structure: each valid possibility should be included once, and each invalid possibility excluded for a stated reason.

A method to try

  1. List the domain and boundaries first: real or integer, positive or unrestricted, strict or inclusive, distinct or possibly equal. Translate a nondegenerate triangle into strict triangle inequalities before counting integer lengths.
  2. Test qualitatively different permitted cases. Try a negative value, zero, a fraction or an equality boundary only when allowed. Record the comparison each case produces; stop once two valid cases show that no single standard relationship holds.
  3. If the relationship seems fixed, prove the sign of A − B. For reciprocals, track the denominator’s sign before multiplying. For a factored product, identify what the known sign of one factor forces about the other.
  4. For restricted selections, build a reversible counting method. Fix forced positions first, then count remaining choices; or remove required gaps and show how to restore them uniquely. A method is complete only if it neither loses valid outcomes nor counts the same outcome twice.
WORKED EXAMPLE

Follow the reasoning.

Real numbers u and v satisfy uv < 0. Compare Quantity A: u² + v² with Quantity B: (u − v)².

  1. Expand Quantity B: (u − v)² = u² − 2uv + v². Subtract Quantity A from Quantity B to get −2uv.
  2. The condition uv < 0 means that −2uv > 0. Thus B − A is always positive, and Quantity B is greater for every allowed pair. No individual numerical test is needed to complete this proof.
  3. The strict condition matters. If the condition had been uv ≤ 0, values with uv = 0 would produce equality, while values with uv < 0 would still make B greater. Under that changed condition, no single one of “A greater”, “B greater” or “equal” would always hold.
  4. For a separate counting example, suppose a nondegenerate triangle has two sides of lengths 5 and 8 and an integer third side t. The inequalities give 3 < t < 13, so t may be any integer from 4 through 12: nine choices. Including 3 or 13 would count degenerate figures.
CHECK YOUR UNDERSTANDING

Think it through first.

x and y are real numbers satisfying x² + y² = 25. Must x + y be greater than 5?

Show the explanation

No; the relationship of x + y to 5 is undetermined.

The allowed pair (3, 4) gives x + y = 7, greater than 5. The allowed pair (−3, 4) gives x + y = 1, smaller than 5. Both satisfy the same squared equation. The absence of a positivity condition is essential: silently adding one would exclude the second case.

Make it your own

When reviewing each companion question, mark the exact condition that made the reasoning work. Label your final support as either a universal argument or contrasting valid examples. For a counting solution, explain why every counted object satisfies the restrictions and why it appears only once.

This lesson is one part of a short starter pathway. It does not cover the full exam syllabus or establish a predicted score.

Official references

These provider and publisher links explain the exam or underlying concepts. Our lessons and questions are original, independent practice.

View 7 references and checked dates

Original lesson and examples · Checked 2026-10-07 · No provider endorsement

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