Question

\(100x < y\)
\(1000x < 2y\)

Column A	Column B
\(1,100x\)	\(y\)

Hint:

In inequality problems, pay close attention to whether variables can be positive, negative, or zero. If the problem doesn't specify, you must consider all cases. Finding just one scenario with conflicting outcomes is enough to prove the answer is (D).

Answer 1

Step 1: Understanding the Concept:
This problem involves a system of two linear inequalities with two variables. We need to use these inequalities to determine the relationship between the expressions in Column A and Column B. Since there are no constraints on \(x\) and \(y\) (e.g., being positive), we must consider all possibilities.
Step 2: Key Formula or Approach:
First, simplify the given inequalities to get the clearest possible bound for \(y\) in terms of \(x\). Then, test different values for \(x\) (positive, negative, and zero) to see if the relationship between Column A and Column B remains constant.
Step 3: Detailed Explanation:
We are given two inequalities:
1) \(100x<y\), which can be written as \(y>100x\).
2) \(1000x<2y\). Dividing by 2 (a positive number), we get \(500x<y\), which can be written as \(y>500x\).
For both inequalities to hold true, \(y\) must be greater than the larger of the two lower bounds.
If \(x>0\), then \(500x>100x\). So, the controlling inequality is \(y>500x\).
If \(x<0\), then \(500x<100x\). So, the controlling inequality is \(y>100x\).
If \(x = 0\), both inequalities become \(y>0\).
Let's test cases based on the sign of \(x\).
Case 1: Let \(x\) be positive (e.g., \(x=1\)).
The condition becomes \(y>500(1)\), so \(y>500\).
Column A: \(1,100x = 1,100(1) = 1,100\).
Column B: \(y\).
We are comparing 1,100 with \(y\). We know \(y\) must be greater than 500.
- If we choose \(y = 600\) (which is>500), then Column B (600)<Column A (1,100).
- If we choose \(y = 1,200\) (which is>500), then Column B (1,200)>Column A (1,100).
Since we get a different relationship depending on the value of \(y\) we choose, the relationship cannot be determined for positive \(x\).
Because we have already found a scenario where the relationship is indeterminate, we can conclude the answer is (D). We do not need to test \(x \le 0\).
Step 4: Final Answer:
The given inequalities only provide a lower bound for \(y\) in terms of \(x\). They do not provide an upper bound. This means \(y\) can be arbitrarily large. When \(x\) is positive, we need to compare \(1,100x\) with a value \(y\) that is only known to be greater than \(500x\). Since \(y\) could be \(600x\) or \(1,200x\), we cannot establish a fixed relationship. Therefore, the relationship cannot be determined.