Quantitative Comparisons: Prove the Relationship

A reliable method

1. List the allowed inputs.
2. Simplify the difference or use a reversible transformation.
3. Look for counterexamples at negative values, fractions, zero, or boundaries when allowed.
4. Choose a definite relationship only when it holds throughout the domain.

Example 1

Compare Quantity A = \(x^{2}\) with Quantity B = \(x\). The real input x is positive. Decide: left greater, right greater, equal, or not determined.

1. Use only the stated input restriction; do not silently assume x is an integer.
2. Subtract B from A: \(x \left(x - 1\right)\). A positive difference means A is greater; a negative one means B is greater.
3. Within the allowed domain, possible outcomes are: left is greater, right is greater, equal. Therefore the comparison is not determined by the given information.

Answer: not determined by the given information

Example 2

Compare Quantity A = \(2 x + 3\) with Quantity B = \(2 x + 1\). The real input x is unrestricted. Decide: left greater, right greater, equal, or not determined.

1. Use only the stated input restriction; do not silently assume x is an integer.
2. Subtract B from A: \(2\). A positive difference means A is greater; a negative one means B is greater.
3. Within the allowed domain, possible outcomes are: left is greater. Therefore the comparison is left is greater.

Answer: left is greater

Common mistakes

• Do not assume a variable is a positive integer.
• Two different valid outcomes mean the relationship is not determined.

Practice without the answer first

1. Compare Quantity A = \(x^{2}\) with Quantity B = \(0\). The real input x is nonzero. Decide: left greater, right greater, equal, or not determined.
2. Compare Quantity A = \(\sqrt{x^{2}}\) with Quantity B = \(x\). The real input x is negative. Decide: left greater, right greater, equal, or not determined.
3. Compare Quantity A = \(\left(x + 1\right)^{2}\) with Quantity B = \(x^{2} + 2 x + 1\). The real input x is unrestricted. Decide: left greater, right greater, equal, or not determined.

These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.