The system of inequalities consists of two distinct parts: y < 0.75x and y > 3x + 2. To determine where the solution set lies, we need to analyze each inequality separately, and then find the region where they overlap on the graph.
1. **Inequality 1:** y < 0.75x
- This inequality represents all the points that lie below the line defined by y = 0.75x. This line has a positive slope of 0.75 and passes through the origin (0,0).
2. **Inequality 2:** y > 3x + 2
- This inequality represents the area above the line defined by y = 3x + 2. This line has a steeper slope of 3 and a y-intercept of 2, which means it crosses the y-axis at (0, 2).
Now, let's graph these two lines to visualize where the solution set is located.
First, plot the two lines:
- The line
y = 0.75xruns from (0,0) with a slope of 0.75. For example, ifx = 4, theny = 3. - The line
y = 3x + 2goes through points such as (0,2) and (1,5).
To then identify the region of overlap:
- For
y < 0.75x, shade below the line. - For
y > 3x + 2, shade above the line.
The solution to the system lies where these shaded areas overlap. This could typically be a region in the first quadrant (depending on the specific intersection points), but confirmation through the equations is needed for exact coordinates.
To find the intersection of the two lines, set the equations equal to each other:
0.75x = 3x + 2. Solving for x provides the intersection point, and evaluating it back into either equation provides the corresponding y. This point will help in precisely identifying the corner of the solution area.
In conclusion, the solution to the system of inequalities lies in the region of the graph where the area below the line y < 0.75x overlaps with the area above the line y > 3x + 2. The precise coordinates of the vertices of this region should be calculated for a complete understanding of the solution set.