Compare quantities: Variables
Prove an algebraic comparison or find allowed values that change the answer.
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Example
Example
Example · Standard
Which relationship holds for every allowed value?
x is a real number.
Roots and bounds · Track both roots, reverse reciprocal order correctly and use fixed-sum identities.
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6 questions · No time limit
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Compare Quantity A and Quantity B for all values allowed by the conditions.
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Reload pageExplore the method
Use real-solution constraints to prove a bound, then test whether equality or a counterexample changes the comparison.
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How this data is usedHelp
Six untimed comparisons use four fixed answers: A is greater, B is greater, both are equal, or the relationship cannot be determined. A determinate answer must hold for every allowed value. Choose cannot be determined when two allowed cases give different relationships, including equality in one case. Easy and Standard rounds cover four reasoning families. Challenge rounds draw six different families from a pool of seven. Reviews give algebraic proofs or exact counterexamples. These are original practice questions, not an official exam or a validated assessment.
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Original practice puzzles, not a standardized assessment.
About this practice
A few matching examples are not a proof. Use every condition to decide whether the relationship always holds.
Learn a method
Prove a comparison or find a counterexampleReport a problem
On by default unless you turn it off. Saved choices are kept. Shares starts, completions, replays and error categories. For ChatGPT visits, starts and completions may include the first QuickenBrain page and country. No answers, drawings or names.
How this data is used