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Use restrictions, algebra and counterexamples to decide a quantitative comparison.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Select the relationship that follows for every permitted case, and recognize when two allowed cases give different relationships.
You need arithmetic with signed numbers and fractions, the distributive law, and the meanings of greater than and equal to. Subtracting the same expression from both quantities preserves their relationship. Multiplying both by a positive number preserves the direction; multiplying by a negative number reverses it. If those rules feel uncertain, practise them before adding a time limit. Speed cannot repair an invalid comparison.
| Term | What it means |
|---|---|
| Domain | All values allowed by the stated conditions. |
| Universal relationship | A comparison that holds in every permitted case. |
| Counterexample | One allowed case that contradicts a proposed universal relationship. |
| Underdetermined | The information allows different relationships between the quantities. |
A quantitative comparison asks for a relationship between two quantities. You may be able to prove that the first is greater, that the second is greater, or that they are equal. The remaining possibility is that the stated information permits different relationships. That last answer does not mean the mathematics is difficult. It means that no single relationship follows from the conditions.
Read the restrictions before calculating. A real number can be negative, zero, a fraction, or an integer. A positive number can still be below one. An integer cannot be one-half. If a variable appears in both columns, it has the same value in both: comparing x + 4 with x + 9 does not permit choosing a different x for each expression. You are comparing a pair under one set of conditions, then deciding whether its relationship persists across every permitted assignment.
Our original practice uses the four comparison choices in a fixed order. Select the statement itself; no calculation is submitted for grading. The feedback supplies an algebraic reason or explicit counterexamples. Correct selection is evidence about this bounded question, not a certification that every step of your private working was valid.
Another way: steps
Read the domain; simplify safely; look for a proof; if needed, test contrasting allowed cases; match the resulting relationship to the answer.
Suppose A is 4x + 11 and B is 4x + 6. Subtract B from A. The variable terms cancel and the difference is five. A positive difference establishes that A is greater for every real x; no substitution is needed. If the difference were zero, the quantities would be equal. If negative, B would be greater. This method is especially helpful when the two columns share most of their structure. It does not require finding the actual value of either column. State the sign of the difference and why that sign cannot change under the restrictions.
Trying x = 2 can reveal a mistake or suggest a relationship, but it cannot normally establish a claim about every positive real number. When powers or reciprocals appear, test a fraction between zero and one as well as a number above one. When signs are unrestricted, include negative values and zero if the expressions are defined there. These choices are a search strategy, not a universal checklist that guarantees a proof. Once two valid cases produce different relationships, however, the search can stop: those two cases already refute every proposed fixed relationship.
You do not need one case with A greater and another with B greater to establish insufficient information. One case with A greater and another with equality is enough. For example, compare the absolute value of a real variable with that variable. A negative input makes the absolute value greater; a positive input makes the quantities equal. It is true that the absolute value is never smaller, but that is not one of the three fixed relationships offered. Read the exact choices rather than replacing them with a weaker statement that happens to be true.
A proposed counterexample must satisfy all the conditions at once. If n is an integer above one, n = one-half is irrelevant even if it reverses your comparison. If x and y are positive and x + y = 10, choosing x = 8 and y = 7 is equally invalid. Write each proposed assignment beside the conditions and check it before evaluating the two columns. In geometry, retain positivity and any stated side or angle relationships. An invalid case tells you nothing about the answer, because the problem never allowed that situation.
Consider two expressions that both contain a factor x. Cancelling x in a comparison is safe only after checking whether it is positive, negative, or zero. Positive multiplication preserves order; negative multiplication reverses it; division by zero is undefined. If the domain permits all three possibilities, separate them or use subtraction instead. The same caution applies when squaring: squaring preserves order for nonnegative quantities, but it can reverse the order of negative quantities or hide a sign difference. Choose transformations because their conditions hold, not because they make the expressions shorter.
Before comparing a measured quantity with a number, identify the units attached to each. Three-quarters of a liter and 750 milliliters represent the same volume. Their written numerals look different because their units differ. Convert once and then compare. A sketch likewise supplies only the information justified by its labels and stated conditions; visual size alone is not a proof of a length or area relationship. You can often construct two figures satisfying the same given measurements but with different areas. Describe those dimensions explicitly so the comparison can be checked without trusting a drawing.
After checking an answer, classify what failed. Did you silently assume a positive variable was an integer? Did you find one example and mistake it for a proof? Did you overlook equality as a competing relationship? Did a unit conversion or sign rule fail? Write one short correction and then solve a fresh case of that kind. Repeating a remembered answer is review, not an unfamiliar assessment. The teaching, practice and exit banks here contain different cases, although a later visit can repeat a case within its finite bank. Use that distinction when judging your progress.
Begin these comparisons without a timer. Before choosing an answer, identify the domain and say whether your evidence is a general proof or a pair of counterexamples. Check the explanations after the attempt. If you were correct by a guess or by an invalid argument, record that honestly rather than treating the selection as secure understanding.
The GRE's Quantitative Reasoning measure includes comparison questions alongside other formats. These items are original instructional practice and do not predict an official scaled score. A timed drill can later use the shared practice player, but this lesson first teaches the reasoning needed to make timing useful.
For a later review, mix comparisons with arithmetic questions that require a numerical answer. The comparison may not require computing either quantity exactly. Notice when cancellation, a sign argument or two contrasting cases saves work. Keep first answers distinct from corrected answers and avoid counting a familiar repeated case as fresh evidence of readiness.
Positive does not mean greater than one. A squared equation can have a negative root. A fixed order of answer choices is not a pattern for guessing. Insufficient information is a claim you can support with two valid cases, not a label for a question you could not solve. Equality versus greater is already enough variation. When using a calculator, keep the mathematical restrictions visible: a decimal result can evaluate a case, but the device does not decide whether that case represents every allowed value.
Read the conditions.
x is real; A = 5x + 8; B = 5x + 3.
The same real x appears in both quantities.
Subtract the quantities.
A - B = 5.
Both copies of 5x cancel without changing the comparison.
Determine the sign.
Five is positive.
The difference is positive for every permitted x.
Select and verify.
Quantity A is greater.
B greater and equality contradict the positive difference; the relationship does not vary, so insufficient information is also wrong.
Read the full domain.
0 < z < 1; A = z; B = z squared.
z is positive but strictly below one.
Factor the difference.
A - B = z(1 - z).
The difference avoids division and keeps the domain visible.
Check both factors.
z > 0 and 1 - z > 0.
Both signs follow from the two parts of the stated interval.
Combine the signs.
z(1 - z) > 0.
The product of two positive numbers is positive throughout the allowed interval.
Reject the competing answers.
Quantity A is greater.
Equality occurs at excluded endpoints, B cannot be greater in this interval, and a universal argument rules out insufficient information.
Read the equation.
w squared = 49; A = w; B = 2.
The equation constrains w without saying it is positive.
Retain both roots.
w = 7 or w = -7.
Both square to 49; neither root can be discarded.
Compare the positive-root case.
At w = 7, A > B.
Seven is greater than two.
Compare the negative-root case.
At w = -7, A < B.
Negative seven is less than two.
Check both assignments against the condition.
Both assignments satisfy the given equation.
The changing relationship is not an artifact of an invalid example.
Select the conclusion.
The relationship cannot be determined.
Each fixed inequality fails in one valid case and equality fails in both; uncertainty follows from the information itself.
Read the expressions.
p > 2; A = p squared; B = 2p.
The boundary two matters because both quantities contain p.
Factor their difference.
A - B = p(p - 2).
Decide the sign of each factor before continuing.
Finish the sign argument.
x is a real number. Quantity A: x + 7 Quantity B: x + 2 Compare the quantities using only the stated information.
y is a real number and y squared equals 25. Quantity A: y Quantity B: 0 Compare the quantities using only the stated information.
n is an integer greater than 1. Quantity A: n squared Quantity B: n Compare the quantities using only the stated information.
0 < t < 1. Quantity A: t squared Quantity B: t Compare the quantities using only the stated information.
x is a real number. Quantity A: 3(x + 2) Quantity B: 3x + 6 Compare the quantities using only the stated information.
x is positive. Quantity A: x squared Quantity B: x Compare the quantities using only the stated information.
a and b are positive, and a > b. Quantity A: 1/a Quantity B: 1/b Compare the quantities using only the stated information.
x is a real number. Quantity A: the absolute value of x Quantity B: x Compare the quantities using only the stated information.
A rectangle has perimeter 24 units. Its side lengths are positive. Quantity A: its area in square units Quantity B: 32 Compare the quantities using only the stated information.
A bottle contains exactly 0.75 liter. Quantity A: its volume in milliliters Quantity B: 750 Compare the quantities using only the stated information.
Two runners move at constant speeds. Runner A travels farther than runner B. Both distances and both travel times are positive, but the travel times are not stated. Quantity A: runner A's speed Quantity B: runner B's speed Compare the quantities using only the stated information.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
r is a negative real number. Quantity A: r cubed Quantity B: 0 Compare the quantities using only the stated information.
Tomorrow, compare x squared with x when x is positive. Recover two cases with different relationships, then explain how the answer changes if x is an integer greater than one. Next, strengthen the arithmetic tools used in these comparisons.
17. Complete a comparison proof, step 3
Both factors are positive, so A is greater.
The strict restriction excludes zero factors and makes the same conclusion hold in every case.