Question

determine whether the following statement is true or false. the graph of 4x - 6y > 3 includes the line 4x - 6y = 3. 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 with the > symbol includes the line, while the ≥ symbol does not include the line. c. the statement is true. the graph of an inequality always includes the line. d. the statement is false. the graph of an inequality with the > symbol does not include the line, while the ≥ symbol includes the line.

Explanation:

Step1: Recall Inequality Graph Rules

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

Step2: Analyze Given Inequality

The inequality is \(4x - 6y>3\). The symbol is \(>\), so the line \(4x - 6y = 3\) is not included in the graph.

Step3: Evaluate the Statement

The statement claims the graph includes the line \(4x - 6y = 3\), which is false. Now check the options:

• Option A: Says the statement is false and explains that an inequality with \(>\) never includes the line (matches our analysis).
• Option B: Incorrectly states \(>\) includes the line and \(\geq\) does not (opposite of rules).
• Option C: Incorrectly says all inequality graphs include the line.
• Option D: Incorrectly swaps the rules for \(>\) and \(\geq\).

Answer:

A. The statement is false. The graph of an inequality never includes the line.