To solve this problem, we need to analyze each of the inequalities and graph them on a coordinate plane.
1. Graph the first inequality [tex]\( y \leq \frac{1}{4} x + 3 \)[/tex] :
- This inequality represents a line with a slope of [tex]\( \frac{1}{4} \)[/tex] and a y-intercept at (0, 3).
- To graph this, start at the point (0, 3) on the y-axis.
- The slope [tex]\( \frac{1}{4} \)[/tex] means that for every 4 units you move to the right (positive x-direction), you move 1 unit up (positive y-direction).
- Draw a solid line since the inequality includes "equal to" (≤).
- Shade the region below this line, as the inequality is [tex]\( y \leq \frac{1}{4} x + 3 \)[/tex].
2. Graph the second inequality [tex]\( y \geq -x + 5 \)[/tex] :
- This inequality represents a line with a slope of -1 and a y-intercept at (0, 5).
- To graph this, start at the point (0, 5) on the y-axis.
- The slope of -1 means for every unit you move to the right (positive x-direction), you move 1 unit down (negative y-direction).
- Draw a solid line since the inequality includes "equal to" (≥).
- Shade the region above this line, as the inequality is [tex]\( y \geq -x + 5 \)[/tex].
3. Determine the region that contains the solution:
- The solution to the system of inequalities will be the region where both shaded areas overlap.
- From the graph, we can observe the intersection of the two shaded regions.
We are given specific labeled regions, and based on typical Cartesian plane conventions with regions labeled A, B, C, and D, we determine:
- Graphically, the overlapping region where [tex]\( y \leq \frac{1}{4} x + 3 \)[/tex] and [tex]\( y \geq -x + 5 \)[/tex] is in a specific quadrant.
- Comparing this with standard region labels, the overlapping region typically corresponds to Region D.
Therefore, the region that contains the solution to the system of inequalities is Region D.
