ORIGINAL WORKED EXAMPLE
Two roots, one requested square
What is x²? (1) x=−5. (2) |x|=5.
- Statement 1 gives 25.
- Statement 2 permits −5 and 5, but both square to 25.
- Each branch fixes the target even though statement 2 does not fix x.
Answer D
GMAT · Data Sufficiency · TOPIC LESSON
Data Sufficiency edge cases and counterexamples
Find the permitted exceptions that distinguish a proof from a plausible guess.
BY THE END, YOU CAN
- Explain unique value versus multiple values
- Explain consistent yes and consistent no both sufficient
- Explain integer-positive-domain restrictions
- Explain zero-negative-fraction counterexamples
- Explain redundant versus independent equations
- Explain underdetermined systems
- Explain irrelevant information
- Explain testing boundary cases
BEFORE YOU START
- Data Sufficiency five-case mapping
- Real numbers, integer parity and elementary equations
Original educational lesson. Work through the examples before revealing the practice answers. This original lesson supports current GMAT study through an editorial skill map. It does not reproduce official questions, predict exam content or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
Original GMAT-style worked examples and practice
All passages, examples and exercises here are original. No official GMAC questions, choices or explanations are reproduced. This independent preparation resource is not affiliated with or endorsed by GMAC. GMAT is a trademark of the Graduate Management Admission Council.
01 / UNDERSTAND & APPLY
Unique values and fixed truth values
For every Data Sufficiency exercise here select exactly one case: A = statement 1 alone suffices, statement 2 alone does not; B = statement 2 alone suffices, statement 1 alone does not; C = together sufficient but neither alone; D = each alone sufficient; E = insufficient even together. Assume all stated conditions are consistent. Test each statement with the stem, never with the other statement smuggled in.
The correct answer is determined by variation in the target. x can vary while x² stays fixed; x² can be fixed while x varies. Likewise, a whole interval of x values may yield the same yes/no answer.
02 / UNDERSTAND & APPLY
The no answer is not insufficiency
A proposition is settled whenever its truth value cannot change. “Not enough to prove yes” is not the definition of insufficient: the data might prove no. Conversely, a single example with answer no does not prove the answer is always no.
Test the boundary itself. Strict comparisons such as > and < treat equality differently from ≥ and ≤. An interval that ends at zero may or may not include zero.
ORIGINAL WORKED EXAMPLE
Boundary creates ambiguity
Is x≥0? (1) x²=0. (2) x≤0.
- Statement 1 forces x=0, so the answer is yes.
- Statement 2 permits x=0 (yes) and x=−1 (no).
- The equality case is decisive.
Answer A
ORIGINAL PRACTICE · 1
Is y>4? (1) y≤4. (2) y=4.
Reveal answer & reasoning
D
Both statements guarantee no, independently. An answer fixed at no is fully sufficient.
03 / UNDERSTAND & APPLY
Use exactly the declared domain
A domain restriction is evidence. Positive integers start at 1; nonnegative integers include 0; real numbers include fractions and negatives. “Number” does not by itself mean positive integer. A short interval can contain one integer but infinitely many reals.
Before searching for examples, note whether variables can coincide, equal zero or be negative. Unless ruled out, such cases are legitimate. Do not select mutually inconsistent values for related variables.
ORIGINAL WORKED EXAMPLE
Integer restriction resolves an interval
What is n, a positive integer? (1) n<3. (2) n>1.
- Statement 1 permits n=1 or 2.
- Statement 2 permits n=2 or 3 and many larger values.
- Together 1<n<3; the only integer is 2. Without the integer restriction this would be insufficient.
Answer C
ORIGINAL PRACTICE · 1
What is x, a real number? (1) 1<x<3. (2) x is an integer.
Reveal answer & reasoning
C
The interval alone contains 1.5 and 2.5. Integer status alone permits many values. Together only x=2 remains.
ORIGINAL PRACTICE · 2
Is x>1? (1) x>0. (2) x²>x.
Reveal answer & reasoning
C
Statement 1 allows 1/2 and 2. Statement 2 is x(x−1)>0, allowing x<0 or x>1. Combining with positivity leaves x>1.
04 / UNDERSTAND & APPLY
Zero, negatives and fractions are purposeful tests
Choose counterexamples to challenge an inference, not randomly. For x²>x, test 0<x<1 as well as x>1. For multiplying inequalities, test a negative multiplier. For division by x, test whether x=0 is possible before dividing.
A witness must obey every active condition. To show a statement is insufficient, two legal cases with differing target answers suffice; a case violating even one condition proves nothing.
ORIGINAL WORKED EXAMPLE
Division would discard a legal solution
What is x? (1) x²=3x. (2) x≥0.
- Factor statement 1: x(x−3)=0, so x=0 or 3. Dividing by x would wrongly lose 0.
- Statement 2 allows 0 and 3, among others.
- Both 0 and 3 satisfy the combined information, so the target remains undetermined.
Answer E
ORIGINAL PRACTICE · 1
Is 1/x positive for nonzero real x? (1) x²>0. (2) x<0.
Reveal answer & reasoning
B
Statement 1 holds for either sign and does not determine the reciprocal’s sign. Statement 2 forces a negative reciprocal and therefore a definite no.
05 / UNDERSTAND & APPLY
Independent-looking equations may be redundant
Two equations provide new information only when one is not logically implied by the other over the allowed domain. Multiplying an equation by a nonzero constant changes its appearance but not its solution set.
Equation counting is a useful warning, not a proof. A sum can be fixed with two unknowns and one equation. Nonlinear equations can have several solutions even with as many equations as unknowns.
ORIGINAL WORKED EXAMPLE
A duplicated relationship
What is p? (1) p+q=11. (2) 3p+3q=33.
- Statement 1 allows (p,q)=(1,10) and (2,9).
- Statement 2 reduces to p+q=11 and allows the same cases.
- Combining duplicates adds no restriction.
Answer E
ORIGINAL PRACTICE · 1
What is u+v? (1) 2u+2v=18. (2) u−v=1.
Reveal answer & reasoning
A
The requested sum is 9 from statement 1. Statement 2 permits sums 1 and 3 from (1,0) and (2,1).
06 / UNDERSTAND & APPLY
Underdetermined variables can still determine the target
Rather than solving every variable, rewrite the target using the supplied relationship. A ratio may cancel an unknown scale; a difference may cancel a common offset. However, cancellation must be valid for the specified domain.
Irrelevant information narrows some aspect of the situation without narrowing the target’s answer. It is not necessarily false or useless in every context, just insufficient for this question.
ORIGINAL PRACTICE · 1
For positive a and b, what is a/b? (1) a=3b. (2) a+b=20.
Reveal answer & reasoning
A
Statement 1 fixes ratio 3. Statement 2 allows (a,b)=(10,10) and (15,5), ratios 1 and 3. Positivity ensures the denominator is nonzero.
ORIGINAL PRACTICE · 2
How many tickets were sold? (1) Each ticket cost $8. (2) Tickets were blue.
Reveal answer & reasoning
E
Neither statement gives revenue, capacity sold or a count. Counts 10 and 20 fit both facts; color is irrelevant to the count.
07 / UNDERSTAND & APPLY
Test the endpoint before declaring a range sufficient
For yes/no questions about a threshold, mark both the allowable interval and the question’s cutoff. If the allowable set lies entirely on one side, the answer is settled. If it includes cases on both sides, it is not.
For integer domains, test the integers remaining after combining inequalities. Do not replace a strict bound with a weak one; n<4 allows 3 but excludes 4.
ORIGINAL PRACTICE · 1
Is n divisible by 4, where n is a positive integer? (1) 3<n<9. (2) n is divisible by 4.
Reveal answer & reasoning
B
Statement 1 permits n=4 and n=5, giving yes and no. Statement 2 directly guarantees yes. Independent sufficiency does not require identifying whether n=4, 8 or another multiple.
08 / UNDERSTAND & APPLY
Separate a failed search from a proof
If you have not found a counterexample, change testing strategy: inspect equality, a negative value, a proper fraction, or a hidden second root. Once a general argument covers the allowed set, stop searching.
For a claim of insufficiency, record explicit legal witnesses. For a claim of sufficiency, record why every allowed case agrees. That pairing makes your answer auditable and catches both overconfidence and unnecessary computation.
KEEP THE BIG IDEAS
Quick revision
- The domain controls which counterexamples are legal.
- Check zero before division and both roots before choosing a value.
- Redundant equations do not add independent restrictions.
- A fixed target can coexist with unfixed variables.
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