Problem 57 - Introductory Algebra for College Students

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

Both graphs are represented by the same line, y = 1.5x - 3, with a slope of 1.5 and y-intercept of -3. The major difference between the graphs is the region representing the solution to these inequalities and whether points on the line are considered part of the solution. In the inequality \(3x - 2y > 6\), points on the line are not included in the solution and the region below the line represents the solution. In the case of \(3x - 2y ≤ 6\), points on the line are considered part of the solution and the region above and on the line represents the solution.

Step 1: Graphing the first inequality

Rearrange the inequality to get y in terms of x. So, \(3x -2y > 6\) becomes \(y < 1.5x - 3\). Graph this inequality. This results in a graph of a line with a slope of 1.5 and y-intercept of -3. Because it is a 'greater than' inequality, shading for the graph will be under the line and the line will be dashed, indicating that points on the line are not included in the solution.

Step 2: Graphing the second inequality

The second inequality can also be rearranged: \(3x - 2y ≤ 6\) becomes \(y ≥ 1.5x - 3\). The resulting line will be identical to that in the first graph, but the shading will be placed over the line, indicating that y is greater than or equal to \(1.5x - 3\). Moreover, because it's a non-strict inequality, the line will be solid, indicating that points on the line are included in the solution.

Step 3: Comparing the graphs

Both graphs will have the same line with a slope of 1.5 and intercept of -3, but the shading of the regions and the line types will be different. In the first equation, the solution is for values less than the line, while in the second equation, the solution includes points on the line and above it. Thus, the major difference is in the inclusion or exclusion of the points on the line, and the region of the graph that represents the solution to the inequality.