Short Answer 
The solution to the system of inequalities (x + 3y > 6) and (y ‚a• 2x + 4) involves graphing the lines corresponding to the inequalities, shading the regions accordingly, and identifying the intersection area. The valid solution region can be expressed as (x > -0.857) and (y ‚a• 2.286), indicating any point within this region satisfies both inequalities.
Step 1: Understanding the Inequalities
To solve the system of inequalities graphically, we need to examine each inequality carefully. We have the inequalities: x + 3y > 6 and y ‚a• 2x + 4. These inequalities define regions on a graph. By isolating y in both inequalities, we can express them in slope-intercept form, which helps in plotting their graphs effectively.
Step 2: Graphing the Inequalities
Next, we will graph the lines represented by each inequality on a coordinate plane. Here’s how to do it:
- For x + 3y = 6, rearrange to y = -1/3x + 2, then plot the line and shade the area above the line to indicate x + 3y > 6.
- For y = 2x + 4, plot the line as well and shade the area above or on the line for y ‚a• 2x + 4.
Step 3: Finding the Solution Region
After graphing, the solution to the system involves identifying where the shaded areas from both inequalities overlap. This intersection represents the region of valid solutions. From the analysis of the graph, we derive the ranges:
- x > -0.857
- y ‚a• 2.286
This means that any point (x, y) within this region satisfies both inequalities.