A reliable method
Simplify a difference or ratio when its sign is known. If no fixed relationship emerges, try allowed positive, negative, zero and fractional values as the conditions permit.
The common trap
Do not select equality because one test value gives equality. To prove an indeterminate relationship, show two allowed assignments that lead to different comparisons.
Make the connection
Keep the four standard choices in view. A counterexample can settle an indeterminate case more efficiently than solving for every possible value.
Contrast the cases
If x²=9, values 3 and −3 make x greater and smaller than zero, respectively. That establishes an indeterminate comparison. Testing x=3 alone would miss a permitted counterexample. If the extra condition x>0 is added, only 3 remains. For x>2, comparing x² with 2x can be proved without trial values: their difference is x(x−2), and both factors are positive. A proof and a counterexample answer different logical tasks.
Transfer and explain
Transfer: with x>0 compare x with 1/x. Values 1/2 and 2 give opposite orderings, so the relationship is not fixed. Add x>1 and explain why x is then always greater.
Let x be any positive real number. Quantity A: x/3. Quantity B: 3/x.
- If x = 3, the quantities are equal.
- If x = 6, Quantity A is 2 and Quantity B is 1/2.
- Both assignments are allowed, so no single relationship is forced.
Compare the other answer choices
Assumes x is greater than the comparison constant, although smaller positive inputs are allowed.
Assumes x is less than the comparison constant, although larger positive inputs are allowed.
Uses equality at x equal to the constant as if it proved equality at every positive input.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference