Problem 65 - 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 two inequalities \(3x - 2y > 6\) and \(3x - 2y \leq 6\) share the same line when graphed. However, \(3x - 2y > 6\) covers the area above the line (not including the line), while \(3x - 2y \leq 6\) includes the line and covers the area below the line.

Step 1: Graph the first inequality

Start by graphing the inequality \(3x - 2y > 6\). It can be helpful to first graph the line as if it was an equation by making it \(3x - 2y = 6\). The graph will be a straight line, where for every 2 units increase in y, there is a decrease of 3 units in x. The inequality portion implies that the area of the graph that satisfies this inequality is the one above the line.

Step 2: Graph the second inequality

Next, graph the second inequality \(3x - 2y \leq 6\). Similar to the first case, start by graphing it as if it was an equation \(3x - 2y = 6\). Again, it's a straight line but this time the inequality indicates that the area of the graph satisfying this inequality is the area below or on the line.

Step 3: Compare the two graphs

Having graphed both inequalities, it can now be observed that they both lie on the same line. The difference comes in the area they cover. The first inequality does not include the line and only covers the area above the line. The second inequality includes the line and the area below the line.