To solve the system of inequalities graphically, we will follow a step-by-step approach:
### Step 1: Graph the boundary lines for each inequality
1. First inequality: [tex]\( y < -\frac{1}{2}x + 4 \)[/tex]
- The boundary line is [tex]\( y = -\frac{1}{2}x + 4 \)[/tex].
- To graph this line, find two points:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = -\frac{1}{2}(0) + 4 = 4 \)[/tex]. This gives the point (0, 4).
- When [tex]\( x = 8 \)[/tex]: [tex]\( y = -\frac{1}{2}(8) + 4 = 0 \)[/tex]. This gives the point (8, 0).
- Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((8, 0)\)[/tex] and draw a dashed line through them, as the inequality is strict ([tex]\(<\)[/tex]).
2. Second inequality: [tex]\( y \geq 3x - 3 \)[/tex]
- The boundary line is [tex]\( y = 3x - 3 \)[/tex].
- To graph this line, find two points:
- When [tex]\( x = 0 \)[/tex]: [tex]\( y = 3(0) - 3 = -3 \)[/tex]. This gives the point (0, -3).
- When [tex]\( x = 1 \)[/tex]: [tex]\( y = 3(1) - 3 = 0 \)[/tex]. This gives the point (1, 0).
- Plot the points [tex]\((0, -3)\)[/tex] and [tex]\((1, 0)\)[/tex] and draw a solid line through them, as the inequality is non-strict ([tex]\(\geq\)[/tex]).
### Step 2: Determine the regions to shade
- For [tex]\( y < -\frac{1}{2}x + 4 \)[/tex], the region below the dashed line should be shaded.
- For [tex]\( y \geq 3x - 3 \)[/tex], the region above the solid line should be shaded.
### Step 3: Find the intersection of the shaded regions
The solution set is where the shaded regions for both inequalities overlap. This section is the region below the dashed line [tex]\( y = -\frac{1}{2}x + 4 \)[/tex] and above the solid line [tex]\( y = 3x - 3 \)[/tex].
### Step 4: State the coordinates of a point in the solution set
To find the coordinates of a point in the solution set:
1. Identifying a point within the overlapping shaded region visually, we might test simple points.
For this particular system, let's test the point [tex]\((1, 1)\)[/tex]:
- Substitute [tex]\((1, 1)\)[/tex] into the first inequality:
[tex]\[
1 < -\frac{1}{2}(1) + 4 = 3.5 \quad \text{(True)}
\][/tex]
- Substitute [tex]\((1, 1)\)[/tex] into the second inequality:
[tex]\[
1 \geq 3(1) - 3 = 0 \quad \text{(True)}
\][/tex]
Since the point [tex]\((1, 1)\)[/tex] satisfies both inequalities, it is a valid point in the solution set.
### Conclusion
We have graphically solved the system of inequalities, and a point in the solution set is [tex]\((1, 1)\)[/tex]. This is verified as it satisfies both inequalities.
