Question

determine whether the following statement is true or false. the graph of 3x - 2y > 4 includes the line 3x - 2y = 4. choose the correct answer below. a. the statement is false. the graph of an inequality never includes the line. b. the statement is true. the graph of an inequality always includes the line. c. the statement is false. the graph of an inequality with the > symbol does not include the line, while the ≥ symbol includes the line. d. the statement is true. the graph of an inequality with the > symbol includes the line, while the ≥ symbol does not include the line.

Explanation:

Step1: Recall inequality graph rules

For a linear inequality, if the symbol is \(>\) or \(<\), the line is dashed (not included). If it's \(\geq\) or \(\leq\), the line is solid (included).

Step2: Analyze \(3x - 2y > 4\)

The inequality has \(>\), so its graph uses a dashed line, meaning the line \(3x - 2y = 4\) is not included. Thus, the statement "The graph of \(3x - 2y > 4\) includes the line \(3x - 2y = 4\)" is false. Now check options:

• Option A: False, but "never includes" is wrong (since \(\geq\)/\(\leq\) do include). Eliminate.
• Option B: Statement is false, so "true" is wrong. Eliminate.
• Option C: Statement is false (line not included for \(>\)), and explanation matches ( \(>\) doesn't include, \(\geq\) does). Correct.
• Option D: Explanation about \(>\) and \(\geq\) is reversed. Eliminate.

Answer:

C. The statement is false. The graph of an inequality with the \(>\) symbol does not include the line, while the \(\geq\) symbol includes the line.