Quantitative Comparison Reasoning - Quantitative Reasoning

1. Identify the QC result: Quantity A: $x+y$; Quantity B: $x$; given $y<0$.

   B. Adding a negative $y$ to $x$ decreases the value, making $x+y < x$.

2. What is the correct QC strategy if you can find two valid cases giving different outcomes?

   Choose D: the relationship cannot be determined. Different outcomes from valid cases indicate the comparison is not consistent, requiring choice D.

3. What are the four Quantitative Comparison answer choices (A–D) in words?

   A: Q1>Q2; B: Q2>Q1; C: equal; D: cannot determine. These options outline the possible comparisons in Quantitative Comparison questions on the GRE.

4. Identify the QC result: Quantity A: $x^2$; Quantity B: $x$; given $x<0$.

   A. For $x<0$, $x^2$ is positive while $x$ is negative, making $x^2$ greater.

5. Identify the QC result: Quantity A: $\frac{x}{|x|}$; Quantity B: $1$; given $x$ is real and $x\ne 0$.

   D. The expression $\frac{x}{|x|}$ equals 1 for positive $x$ and -1 for negative $x$, varying relative to 1.

6. Identify the QC result: Quantity A: $x-y$; Quantity B: $x$; given $y>0$.

   B. Subtracting a positive $y$ from $x$ decreases the value, making $x-y < x$.

7. Identify the QC result: Quantity A: $x^2$; Quantity B: $x$; given $x>1$.

   A. For $x>1$, $x^2$ exceeds $x$ since squaring a number greater than 1 increases its value.

8. What QC conclusion is valid if you prove $Q_1-Q_2=0$ for all allowed values?

   Choose C: the two quantities are equal. A zero difference for all values confirms the quantities are identical.

9. Identify the QC result: Quantity A: $x+y$; Quantity B: $x$; given $y>0$.

   A. Adding a positive $y$ to $x$ increases the value, making $x+y > x$.

10. What QC conclusion is valid if you prove $Q_1-Q_2<0$ for all allowed values?

    Choose B: $Q_2$ is greater. A negative difference for all values confirms Quantity 2 exceeds Quantity 1 consistently.

11. What is the key caution about taking reciprocals when quantities may be negative or zero?

    Reciprocal reverses order for positives; sign/zero require cases. Taking reciprocals reverses inequalities for positives but necessitates cases for negatives or zero.

12. Identify the QC result: Quantity A: $\sqrt{x^2}$; Quantity B: $x$; given $x$ is real.

    D. The square root of $x^2$ is $|x|$, which equals $x$ for non-negative $x$ but exceeds $x$ for negative $x$.

13. Identify the QC result: Quantity A: $\frac{1}{x}$; Quantity B: $\frac{1}{y}$; given $0<x<y$.

    A. For $0<x<y$, the reciprocal inequality reverses, so $\frac{1}{x} > \frac{1}{y}$.

14. What is the key caution about canceling a common factor that might be $0$ (for example canceling $x$)?

    Cancel only if the factor is guaranteed nonzero. Canceling a factor risks division by zero unless the factor is proven nonzero.

15. Identify the QC result: Quantity A: $|x|$; Quantity B: $x$; given $x\ge 0$.

    C. For $x \ge 0$, $|x|$ equals $x$ by definition of absolute value.

16. Identify the QC result: Quantity A: $|x|$; Quantity B: $-x$; given $x\le 0$.

    C. For $x \le 0$, $|x|$ equals $-x$ since $x$ is non-positive.

17. Identify the QC result: Quantity A: $x^2$; Quantity B: $x$; given $0<x<1$.

    B. For $0<x<1$, $x^2$ is less than $x$ as squaring a fraction yields a smaller result.

18. What is the key caution about squaring both sides when sign is unknown (for example comparing $a$ and $b$)?

    Squaring can change order when negatives are possible. Squaring both sides preserves order for positives but may reverse it if negatives are involved.

19. What QC conclusion is valid if you prove $Q_1-Q_2>0$ for all allowed values?

    Choose A: $Q_1$ is greater. A positive difference for all values confirms Quantity 1 exceeds Quantity 2 consistently.

20. What is the key caution about multiplying or dividing an inequality by an unknown-sign variable $x$?

    You cannot do it without cases; the inequality may flip. Multiplying or dividing by a variable of unknown sign requires case analysis to avoid inequality reversal.

21. Identify the QC result: Quantity A: $\frac{x^2}{x}$; Quantity B: $x$; given $x\ne 0$.

    C. Simplifying $\frac{x^2}{x}$ yields $x$ for $x \ne 0$, confirming equality.

22. Identify the QC result: Quantity A: $x^3$; Quantity B: $x^2$; given $0<x<1$.

    B. For $0<x<1$, multiplying $x^2$ by $x<1$ yields $x^3 < x^2$.

23. What is the key caution about comparing expressions involving absolute value, such as $|x|$?

    $|x|$ depends on sign; split into cases $x\ge 0$ and $x<0$. Absolute value expressions require case splitting based on the sign to accurately compare.

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