Question

A student wrote the inequality: y > 4x - 17. Her friend solved the inequality and said that (12, 31) is one of the possible solutions of this inequality. Is her friend correct?

Answer 1

ok, why is she wrong

Answer 2

To determine if the point \( (12, 31) \) is a solution to the inequality \( y > 4x - 17 \), we can substitute \( x = 12 \) and \( y = 31 \) into the inequality and check if it holds true.

First, we calculate \( 4x - 17 \) when \( x = 12 \):

\[
4(12) - 17 = 48 - 17 = 31
\]

Now, we substitute this value into the inequality:

\[
y > 4x - 17 \implies 31 > 31
\]

The statement \( 31 > 31 \) is false. Therefore, the point \( (12, 31) \) does not satisfy the inequality \( y > 4x - 17 \).

Thus, the friend's claim is incorrect.

Answer 3

The friend's statement that \( (12, 31) \) is a solution to the inequality \( y > 4x - 17 \) is incorrect because when we substitute the coordinates \( x = 12 \) and \( y = 31 \) into the inequality, we find that it does not satisfy the required condition of the inequality.

Here's the step-by-step reasoning:

1. **Plug in the \( x \) value**: We substitue \( x = 12 \) into the expression on the right side of the inequality:
\[
4(12) - 17 = 48 - 17 = 31
\]

2. **Analyze the inequality**: The original inequality is:
\[
y > 4x - 17
\]
Thus, substituting the calculated value, we have:
\[
y > 31
\]

3. **Substitute the \( y \) value**: Next, we check the \( y \) value of the point \( (12, 31) \):
\[
31 > 31
\]

4. **Determine the validity**: The statement \( 31 > 31 \) is **not true**. In fact, it is equal, which means that \( 31 \) is not greater than \( 31 \).

Since the point \( (12, 31) \) does not satisfy the condition of the inequality \( y > 4x - 17 \), we conclude that the friend's assertion was incorrect. A point must satisfy the strict inequality (i.e., \( y \) must be greater than \( 31 \)) for it to be considered a solution to the inequality.