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

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 2

ok, why is she wrong

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.