Problem 92 - College Algebra Essentials

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 two inequalities have the same boundary line \(y = \frac{3}{2}x - 3\), but one is dashed (for \(3x - 2y > 6\)) and the other is solid (for \(3x - 2y \leq 6\)). The solid line indicates that the points on the line are solutions for the second inequality while they are not for the first. Also, the solution to the first inequality \(3x - 2y > 6\) is above the line, whereas the solution for the second inequality \(3x - 2y \leq 6\) is below the line.

Step 1: Graphing the first Function

Start by graphing the first inequality \(3x - 2y > 6\). Begin by treating the inequality as an equation i.e. \(3x -2y = 6\). That can be rewritten as \(y = \frac{3}{2}x - 3\). Drawing this on the graph, it will be a dashed line since the inequality does not include equals to. The shaded area will be above the line as this is a 'greater than' inequality.

Step 2: Graphing the second Function

Following a similar method for the second inequality \(3x - 2y \leq 6\). Treating it as an equation and rewriting will give us \(y = \frac{3}{2}x - 3\). Drawing this on the graph will be a solid line as the inequality includes an equals to sign. The area below this line will be shaded as this is a 'lesser than or equal to' inequality.

Step 3: Discussing Similarities and Differences

Now comparing the two inequalities, it can be seen that they have the exact same boundary line \(y = \frac{3}{2}x - 3\), but one is dashed (for the first one) and the other is solid (for the second one). The solid line indicates that the points on the line are solutions for the second inequality while in the first case they are not. The shaded areas, representing the solutions of the inequalities, are on different sides of the boundary line, above in the first case and below in the second.