Question

determine whether the following statement is true or false. when graphing $3x - 2y \geq 5$, to determine which side of the line to shade, choose a test point on $3x - 2y = 5$. choose the correct answer below. a. the statement is true. it does not matter where the test point is chosen. b. the statement is false. the test point should always be chosen to match the inequality. c. the statement is false. the test point should be chosen from one of the half - planes. d. the statement is true. the test point should always be on the line.

Explanation:

Step1: Recall Inequality Graphing

When graphing a linear inequality \( ax + by \geq c \) (or other forms), the line \( ax + by = c \) is the boundary. Test points are used to determine which half - plane to shade. A test point on the line \( ax+by = c \) will satisfy \( ax + by=c \), but for the inequality \( ax + by\geq c \), we need to test points from the regions (half - planes) that are not on the line to see if they satisfy the inequality. If we choose a test point on the line \( 3x - 2y=5 \), plugging it into the inequality \( 3x - 2y\geq5 \) will give \( 5\geq5 \), which is true, but this point is on the boundary. To determine the correct half - plane to shade, we should choose test points from the half - planes (regions) that are not on the line. So the statement "When graphing \( 3x - 2y\geq5 \), to determine which side of the line to shade, choose a test point on \( 3x - 2y = 5 \)" is false. Also, test points should be chosen from one of the half - planes (the regions above or below the line, not on the line) to see if they satisfy the inequality. So option C is correct because the statement is false and the test point should be chosen from one of the half - planes.

Step2: Analyze Each Option

• Option A: The statement is false (as test points on the line don't help in determining the half - plane to shade), so A is wrong.
• Option B: Test points are not chosen to "match the inequality" in the way this option suggests. We choose test points from the half - planes to see if they satisfy the inequality, so B is wrong.
• Option C: The statement is false (test points on the line don't help in finding the half - plane) and test points should be chosen from one of the half - planes, so C is correct.
• Option D: The statement is false (test points on the line are not used to determine the half - plane to shade), so D is wrong.

Answer:

C. The statement is false. The test point should be chosen from one of the half - planes.