Problem 92 - Algebra and Trigonometry

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 show the same trend with an equivalent slope and y-intercept. The main difference is their boundary; \(3x - 2y > 6\) is represented by a dotted line and doesn't include points on the line itself. However, \(3x - 2y \leq 6\) is represented by a solid line and includes points on the line itself, hence the values that satisfy this inequality are shaded below the line including the line itself.

Step 1: Convert both inequalities into the slope-intercept form

The slope-intercept form of an equation is \(y = mx + b\), where m is the slope and b is the y-intercept. Convert \(3x - 2y > 6\) to \(y < (3/2)x - 3\) and \(3x - 2y \leq 6\) to \(y \leq (3/2)x - 3\). These forms are easier to graph as the slope and y-intercept can be directly identified.

Step 2: Graphing the inequalities

For \(y < (3/2)x - 3\), graph a dotted line (which means the line is not included in the solution) with slope \(3/2\) and y-intercept \(-3\). For \(y \leq (3/2)x - 3\), graph a solid line (which means the line is included in the solution) with the same slope and y-intercept.

Step 3: Shading the regions

For the inequality \(y < (3/2)x - 3\), shade the regions below the line as it represents values less than the line. For the inequality \(y \leq (3/2)x - 3\), also shade the regions below the line and the line itself, as it represents values less than or equal to the line.

Step 4: Comparing the graphs

Now, observe the similarities and differences between the two graphs. The lines for both the inequalities have the same slope (gradient) and y-intercept, thus indicating the same trend. The main difference is in the inclusivity of the values on the line - the first inequality excludes the values lying on the line (dotted line), while the second inequality includes them (solid line).