Question
A student wrote the inequality . Her friend solved the inequality and said that (2, 1) is one of the possible solutions of this inequality. Is her friend correct? Responses Yes, because 1 is equal to Yes, because 1 is equal to 2 times 2 minus 3 No, because 1 is greater than . No, because 1 is greater than 2 times 2 minus 3. Yes, because 1 is greater than . Yes, because 1 is greater than 2 times 2 minus 3. No, because 1 is equal to .
Answers
GPT-4o mini
To determine whether the friend's statement is correct, we need to clarify what the inequality is. Although you haven't provided the specific inequality, let's analyze one potential form.
Suppose the inequality is something like \(y < 2x - 3\).
To check if (2, 1) is a solution, we will substitute \(x = 2\) and \(y = 1\) into the inequality:
1. Calculate \(2x - 3\) when \(x = 2\):
\[
2(2) - 3 = 4 - 3 = 1
\]
2. Now check the inequality \(1 < 1\). This is false.
So, in this case, (2, 1) would **not** be a solution.
However, if the inequality were different (for example, \(y \leq 2x - 3\) or \(y > 2x - 3\)), the evaluation might yield different results.
Given the provided responses, let’s break down one appropriate conclusion:
1. "No, because 1 is greater than \(2 \times 2 - 3\)." - This response could be applicable if the inequality in question were \(y < 2x - 3\).
2. "Yes, because 1 is equal to..." - This response could be applicable if the inequality were of the form \(y \geq 2x - 3\).
Without knowing the specific inequality, we can't confirm or deny definitively, but based on the analysis of the possible inequality provided, it seems that "No, because 1 is greater than \(2 \times 2 - 3\)" could be a correct response if that inequality was \(y < 2x - 3\).
If you can provide the specific inequality your friend was working with, I can give more accurate feedback.
Suppose the inequality is something like \(y < 2x - 3\).
To check if (2, 1) is a solution, we will substitute \(x = 2\) and \(y = 1\) into the inequality:
1. Calculate \(2x - 3\) when \(x = 2\):
\[
2(2) - 3 = 4 - 3 = 1
\]
2. Now check the inequality \(1 < 1\). This is false.
So, in this case, (2, 1) would **not** be a solution.
However, if the inequality were different (for example, \(y \leq 2x - 3\) or \(y > 2x - 3\)), the evaluation might yield different results.
Given the provided responses, let’s break down one appropriate conclusion:
1. "No, because 1 is greater than \(2 \times 2 - 3\)." - This response could be applicable if the inequality in question were \(y < 2x - 3\).
2. "Yes, because 1 is equal to..." - This response could be applicable if the inequality were of the form \(y \geq 2x - 3\).
Without knowing the specific inequality, we can't confirm or deny definitively, but based on the analysis of the possible inequality provided, it seems that "No, because 1 is greater than \(2 \times 2 - 3\)" could be a correct response if that inequality was \(y < 2x - 3\).
If you can provide the specific inequality your friend was working with, I can give more accurate feedback.