Write down what values are allowed
Before comparing, identify the domain: real number, integer, positive, nonzero, or interval. A counterexample is valid only if it meets every condition. If x > 0, zero is not allowed; if x is an integer, 1/2 is not allowed. The format does not always ask for a unique value: you can know the relationship between A and B without knowing their values.
For example, compare x + 9 and x + 4. Their difference is 5 for every real x, so A is always greater. By contrast, comparing x² and x requires you to check the domain. Positive integers and positive real numbers allow different possibilities. This approach keeps you from treating every question as an equation to solve.
Short proofs: difference, bound, and equality
Here, choices A, B, C, and D have the meanings given above. These practice exercises are provided by Upsilar. First look for a difference, factorization, or bound.
x is a real number. Quantity A: 3(x + 4). Quantity B: 3x + 7. Compare the quantities.
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The solution, step by step
A.
The method in action
- A − B = 3x + 12 − 3x − 7 = 5. This positive difference proves the answer for every x.
- B reverses the comparison. C overlooks the difference between 12 and 7. D would be wrong: the presence of a variable does not rule out a constant relationship.
t is a real number. Quantity A: 0. Quantity B: (t − 2)² + 1. Compare the quantities.
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The solution, step by step
B.
The method in action
- A square is at least zero, so B is at least 1 and strictly greater than A.
- Testing t = 2 finds the minimum of B, not equality with zero. A and C are impossible; D is unsupported because the bound applies to every real number.
x is a real number and x ≠ 2. Quantity A: (x² − 4)/(x − 2). Quantity B: x + 2. Compare the quantities.
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The solution, step by step
C.
The method in action
- Factoring x² − 4 gives (x − 2)(x + 2). Since x ≠ 2, canceling x − 2 is allowed, and A = x + 2 = B.
- Neither A nor B can be strictly greater; D would overlook that the restriction excludes precisely the case in which the quotient is undefined.
Two counterexamples: vary the relationship
To prove D, two permitted cases with different relationships are enough. You do not need to produce all three possible relationships. By contrast, ten examples showing the same relationship do not always establish a general proof.
x > 0. Quantity A: x². Quantity B: x. Compare the quantities.
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The solution, step by step
D.
The method in action
- When x = 2, A = 4 and B = 2, so A is greater. When x = 1/2, A = 1/4 and B = 1/2, so B is greater.
- Both cases satisfy x > 0. Equality occurs when x = 1, but not for every permitted x; neither A nor B holds in every case. This proves the relationship cannot be determined.
n is an integer and 1 ≤ n ≤ 4. Quantity A: n(5 − n). Quantity B: 6. Compare the quantities.
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The solution, step by step
D.
The method in action
- When n = 1, A is 4, so B is greater. When n = 2, A is 6, so the quantities are equal. That is enough to choose D.
- When n = 3, A is again 6; when n = 4, A is 4. A is never greater, but that does not make B correct: B claims that it is always greater, not that it is greater than or equal.
The pitfall
The pitfall to avoid
Choosing D because you cannot solve the problem is a mistake. D is a mathematical conclusion, not a statement of personal uncertainty. It requires admissible cases that rule out a single relationship; an incomplete calculation does not provide them.
Choose between a proof and test cases
Start by simplifying what can be simplified without additional assumptions. Subtracting the same expression from both quantities preserves the comparison; dividing by a quantity of unknown sign requires separate cases: if it is positive, the inequality direction is preserved; if negative, it is reversed; if zero, division is not allowed. Do not treat an inequality as an equality.
If variables are still undetermined, test qualitatively different values permitted by the domain: a fraction between zero and one, an included boundary, a negative number, or a special equality case. Once two different relationships appear, stop calculating. If they do not, look for what enforces the relationship you observed.
During review, write one sentence proving your answer or record your two counterexamples. A correct guess without a justification deserves the same review as an error. The next question will use different numbers; only the logic behind your decision will carry over.
Key takeaways
- The constraints take priority over the values you would like to test.
- A stable relationship requires proof.
- Two different relationships in two permitted cases are enough to choose D.