Question: Given the system of inequalities and the corresponding graph below, which section of the graph represents all of the solutions to the system?
y \geq x + 4
y \geq - \frac{3}{4} x - 2
y
x
Section A
Section B
Section C
Section D
(A) Section A
Graph: coordinate plane with two solid boundary lines, one rising line y = x + 4 and one falling line y = -3/4 x - 2, dividing the plane into four labeled regions. The solution set is the overlapping shaded region above both lines, located in the top-left area labeled Section A. position_hint=top-left
1. **State the problem:** We need to find which section of the graph represents all solutions to the system of inequalities:
$$y \geq x + 4$$
$$y \geq - \frac{3}{4} x - 2$$
2. **Understand the inequalities:**
- The first inequality, $$y \geq x + 4$$, means the solution includes all points on or above the line $$y = x + 4$$.
- The second inequality, $$y \geq - \frac{3}{4} x - 2$$, means the solution includes all points on or above the line $$y = - \frac{3}{4} x - 2$$.
3. **Graph interpretation:**
- Both lines are solid, indicating points on the lines satisfy the inequalities.
- The solution to the system is the intersection of the regions above both lines.
4. **Identify the overlapping region:**
- The region above $$y = x + 4$$ is the area above the rising line.
- The region above $$y = - \frac{3}{4} x - 2$$ is the area above the falling line.
- The overlapping shaded region above both lines is in the top-left area labeled Section A.
5. **Conclusion:**
- The section representing all solutions to the system is **Section A**.
**Final answer:** Section A