To solve the system of inequalities graphically and find the region that satisfies both, we need to follow these steps:
### Step-by-Step Solution
#### Step 1: Graphing the Inequalities
1. Graph the inequality [tex]\(2x + 5y < -10\)[/tex]:
- First, rewrite the inequality as an equation to find the boundary line:
[tex]\[
2x + 5y = -10
\][/tex]
- Find the intercepts to plot the line.
- x-intercept: Set [tex]\(y = 0\)[/tex]:
[tex]\[
2x = -10 \implies x = -5
\][/tex]
So, the x-intercept is [tex]\((-5, 0)\)[/tex].
- y-intercept: Set [tex]\(x = 0\)[/tex]:
[tex]\[
5y = -10 \implies y = -2
\][/tex]
So, the y-intercept is [tex]\((0, -2)\)[/tex].
- Plot the line that passes through these points [tex]\((-5, 0)\)[/tex] and [tex]\((0, -2)\)[/tex].
- Since the inequality is [tex]\(2x + 5y < -10\)[/tex], shade the region below the line.
2. Graph the inequality [tex]\(x \geq 4\)[/tex]:
- First, rewrite the inequality as an equation to find the boundary line:
[tex]\[
x = 4
\][/tex]
- This is a vertical line passing through [tex]\(x = 4\)[/tex].
- Since the inequality is [tex]\(x \geq 4\)[/tex], shade the region to the right of the line.
#### Step 2: Finding the Shared Region
- The solution to the system is where the shaded regions from both inequalities overlap.
- We have:
- The region below the line [tex]\(2x + 5y = -10\)[/tex].
- The region to the right of the line [tex]\(x = 4\)[/tex].
#### Step 3: Identifying the Shaded Region on the Graph
- Identify the Region:
- Plot the vertical line [tex]\(x = 4\)[/tex].
- Plot the line [tex]\(2x + 5y = -10\)[/tex] as discussed.
- Mark the area below the line [tex]\(2x + 5y = -10\)[/tex].
- Mark the area to the right of [tex]\(x = 4\)[/tex].
- Find the Overlapping Region:
- The overlapping region is to the right of [tex]\(x = 4\)[/tex] and below [tex]\(2x + 5y = -10\)[/tex].
### Conclusion:
- The solution region satisfies [tex]\(2x + 5y < -10\)[/tex] and [tex]\(x \geq 4\)[/tex].
- Graphically, this region should be identifiable after plotting both inequalities.
- The correct answer is the region that simultaneously satisfies both conditions, typically shaded on the graph.
Thus, the specific region we need to shade is determined by the intersection of the two shaded areas described above. This should be the region 'D'.
