Problem 92 - College Algebra

Question

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\) Discuss similarities and differences between the graphs.

Step-by-Step Solution

Answer

The graphs of the inequalities \(3x - 2y > 6\) and \(3x -2y \leq 6\) have the same boundary line and shading direction for the solution set. The major difference between the two is that the boundary line of the first inequality is not included in the solution (shown with a dashed line), while the boundary line of the second inequality is included in its solution set (shown with a solid line).

Step 1: Rearrange the inequalities

First, both the inequalities need to be rearranged to the standard format, y = mx + c. The inequalities \(3x - 2y > 6\) and \(3x -2y \leq 6\) can be rearranged to \(y < \frac{3}{2}x - 3\) and \(y \leq \frac{3}{2}x - 3\)

Step 2: Draw the boundary lines

Next, the lines \(y = \frac{3}{2}x - 3\) needs to be graphed. These are the boundary lines. Use different colors to differentiate the two lines. Remember that for '\(<\)' or \(>\) the lines are dashed (not solid) as they do not include the solution, and for '\(\leq\)' or \(≥\), the line is a solid line as they include the solution.

Step 3: Shade the regions

Then, choose a test point not on the line, such as the origin (0,0), and check which inequality it satisfies. If the test point satisfies the inequality, shade the region that includes the test point. Otherwise, shade the opposite region. The inequality \(y < \frac{3}{2}x - 3\) will have the region below the line shaded, while the inequality \(y \leq \frac{3}{2}x - 3\) will have the region below the line including the line itself shaded.

Step 4: Observe the similarities and differences

Finally, visual comparison of the two graphs shows similarities and differences. The boundary line is the same for both inequalities. The difference lies in whether the line is included in the solution set or not (solid line versus dashed line), and the type of inequality symbol used ('>' versus '\(\leq\)')