Question

determine which point is a solution to the system.
a b c d
e f
yes, it is a solution no, it is not a solution

Explanation:

Step1: Recall System Solution Definition

A solution to a system of inequalities (represented by the shaded region) is a point that lies within the shaded area or on the boundary lines (if included).

Step2: Analyze Each Point

- **Point A**: Visually, it lies within the shaded region. So it should be a solution.
- **Point B**: Lies outside the shaded region (above the solid line and outside the dashed line's shaded side). Not a solution.
- **Point C**: Lies outside the shaded region (above the dashed line and outside the solid line's shaded side). Not a solution.
- **Point D**: On the boundary (solid line), but check the shaded region. The shaded region includes the area below the dashed line and above the solid line? Wait, no—visually, the shaded region is below the dashed line (since dashed line is going up, shaded below) and above the solid line (solid line going up, shaded above? Wait, no, the shaded region is the overlapping area. Wait, looking at the graph, the shaded region is below the dashed line (dashed line has arrow up, shaded below) and above the solid line (solid line has arrow up, shaded above? Wait, no, the shaded area is the gray region. Point A is in gray, so:
- Point A: In shaded region → Yes.
- Point B: Not in shaded → No.
- Point C: Not in shaded → No.
- Point D: On the solid line, but is the solid line part of the solution? The solid line is included (since it's solid), but does D lie in the shaded region? The shaded region is below the dashed line and above the solid line? Wait, the gray area is between the two lines? Wait, no, the dashed line is going from (0,-2) up, and the solid line from (4,0) up. The shaded region is below the dashed line (since the dashed line's arrow is up, and the shaded is below) and above the solid line (solid line's arrow is up, shaded above). Wait, Point A is in the gray, so:
- Point E: Let's check coordinates. Point E: (-3, -3)? Wait, no, the grid: each square is 1 unit. Point E: x=-3, y=-3? Wait, no, the solid line passes through (0,-2) and (4,0), slope 0.5. The dashed line passes through (0,-2) and (2,4), slope 3. The shaded region is below the dashed line (since the dashed line's inequality is y < 3x - 2? Wait, no, when x=0, y=-2, and the dashed line goes up, so if the shaded is below, then y < 3x - 2? But the solid line: y = 0.5x - 2? Wait, no, solid line passes through (4,0) and (0,-2), so slope (0 - (-2))/(4 - 0) = 0.5, so y = 0.5x - 2. The dashed line: passes through (0,-2) and (2,4), slope (4 - (-2))/(2 - 0) = 3, so y = 3x - 2. The shaded region is where y ≥ 0.5x - 2 (since solid line, included) and y ≤ 3x - 2 (dashed line, not included)? Wait, no, the shaded is below the dashed line (y ≤ 3x - 2) and above the solid line (y ≥ 0.5x - 2). So:
- Point A: Let's say coordinates. Point A: x=-6, y=-6? Wait, no, looking at the graph, Point A is at x=-6, y=-6? No, the grid: each square is 1. Point A: x=-6, y=-5? Wait, no, the gray area is on the left. Point A is in the gray, so:
- Point A: satisfies y ≥ 0.5x - 2 and y ≤ 3x - 2? Let's check x=-6: y=-5. 0.5*(-6) - 2 = -3 - 2 = -5. So y = -5, which is equal to 0.5x - 2, so y ≥ 0.5x - 2 (since solid line, included). Then 3x - 2 = 3*(-6) - 2 = -20, so y=-5 ≤ -20? No, that can't be. Wait, maybe I got the inequalities reversed. Maybe the dashed line is y > 3x - 2 (since dashed, not included) and the solid line is y ≤ 0.5x - 2? No, that doesn't make sense. Alternatively, the shaded region is above the solid line and below the dashed line. Wait, when x=0, solid line is y=-2, dashed line is y=-2? No, dashed line at x=2 is y=4, solid line at x=2 is y=0.5*2 - 2 = -1. So at x=2, dashed line y=4, solid line y=-1. The shaded region is between them? No, the gray area is on the left, below the dashed line and above the solid line. Wait, maybe the correct way is to check if the point is in the shaded region (gray). Point A is in gray, so:

Step3: Final Check

- **Point A**: In shaded region → Yes, it is a solution.
- **Point B**: Not in shaded → No.
- **Point C**: Not in shaded → No.
- **Point D**: On the solid line, but is D in the shaded region? The shaded region is below the dashed line and above the solid line. At x=0, y=-2 (Point D). The dashed line at x=0 is y=-2, so y ≤ 3x - 2 at x=0 is y ≤ -2, and y ≥ 0.5x - 2 at x=0 is y ≥ -2. So Point D: y=-2, which satisfies both (y ≥ -2 and y ≤ -2), so it's on the boundary. But is the solid line included (yes, solid) and dashed line? Dashed line is not included, but at x=0, dashed line is y=-2, so y ≤ -2 (dashed line's inequality) and y ≥ -2 (solid line's inequality). So Point D: y=-2, which is equal to both, so it's a solution? Wait, but visually, Point D is on the solid line, and the shaded region includes the area above the solid line and below the dashed line. At x=0, the solid line is y=-2, dashed line is y=-2 (wait, no, dashed line at x=2 is y=4, so at x=0, dashed line is y=3*0 - 2 = -2. So dashed line and solid line intersect at (0,-2) (Point D). So the system is:
- Solid line: y ≥ 0.5x - 2 (since shaded above)
- Dashed line: y < 3x - 2 (since shaded below, dashed line not included)
So at Point D (0,-2):
- y = -2, 0.5*0 - 2 = -2, so y ≥ -2: yes (equal)
- y = -2, 3*0 - 2 = -2, so y < -2: no (equal, but dashed line is not included, so y < 3x - 2 means y ≤ -2? No, dashed line is y = 3x - 2, and the shaded is below, so y < 3x - 2 (since dashed, not included). So at x=0, y=-2 is not less than 3x - 2 (-2), so y=-2 is not < -2. So Point D does not satisfy the dashed line's inequality (y < 3x - 2), so it's not a solution. Wait, this is confusing. But visually, the shaded region is the gray area, which includes Point A, so:

Answer (for each point, assuming we check Point A first):

- **Point A**: Yes, it is a solution
- **Point B**: No, it is not a solution
- **Point C**: No, it is not a solution
- **Point D**: No, it is not a solution
- **Point E**: Let's check. Point E: x=-3, y=-3. Solid line: y=0.5*(-3) - 2 = -1.5 - 2 = -3.5. So y=-3 ≥ -3.5 (yes). Dashed line: y=3*(-3) - 2 = -9 - 2 = -11. So y=-3 < -11? No. Wait, no, maybe the inequalities are reversed. Maybe the dashed line is y > 3x - 2 (shaded above) and solid line y ≤ 0.5x - 2 (shaded below). Then Point A: x=-6, y=-5. Solid line: 0.5*(-6) - 2 = -5, so y=-5 ≤ -5 (yes). Dashed line: 3*(-6) - 2 = -20, so y=-5 > -20 (yes). So Point A satisfies both: y ≤ 0.5x - 2 and y > 3x - 2. Then:
- Point A: Yes
- Point B: x=4, y=1. Solid line: 0.5*4 - 2 = 0, so y=1 ≤ 0? No. So no.
- Point C: x=-5, y=5. Solid line: 0.5*(-5) - 2 = -4.5, so y=5 ≤ -4.5? No.
- Point D: x=0, y=-2. Solid line: 0.5*0 - 2 = -2, so y=-2 ≤ -2 (yes). Dashed line: 3*0 - 2 = -2, so y=-2 > -2? No (equal, dashed line not included). So no.
- Point E: x=-3, y=-3. Solid line: 0.5*(-3) - 2 = -3.5, so y=-3 ≤ -3.5? No. Wait, this is conflicting. The key is visual: the shaded region is the gray area. Point A is in gray, so it's a solution. So the answer for Point A is "Yes, it is a solution", others (B, C, D, E, F) are "No" except A.

(Assuming the question is to check each point, here's the answer for Point A as an example, but since the problem says "which point", likely Point A is the solution.)

Final Answer (for Point A):
Yes, it is a solution

Answer:

(for each point, assuming we check Point A first):

• Point A: Yes, it is a solution
• Point B: No, it is not a solution
• Point C: No, it is not a solution
• Point D: No, it is not a solution
• Point E: Let's check. Point E: x=-3, y=-3. Solid line: y=0.5(-3) - 2 = -1.5 - 2 = -3.5. So y=-3 ≥ -3.5 (yes). Dashed line: y=3(-3) - 2 = -9 - 2 = -11. So y=-3 < -11? No. Wait, no, maybe the inequalities are reversed. Maybe the dashed line is y > 3x - 2 (shaded above) and solid line y ≤ 0.5x - 2 (shaded below). Then Point A: x=-6, y=-5. Solid line: 0.5(-6) - 2 = -5, so y=-5 ≤ -5 (yes). Dashed line: 3(-6) - 2 = -20, so y=-5 > -20 (yes). So Point A satisfies both: y ≤ 0.5x - 2 and y > 3x - 2. Then:
• Point A: Yes
• Point B: x=4, y=1. Solid line: 0.5*4 - 2 = 0, so y=1 ≤ 0? No. So no.
• Point C: x=-5, y=5. Solid line: 0.5*(-5) - 2 = -4.5, so y=5 ≤ -4.5? No.
• Point D: x=0, y=-2. Solid line: 0.50 - 2 = -2, so y=-2 ≤ -2 (yes). Dashed line: 30 - 2 = -2, so y=-2 > -2? No (equal, dashed line not included). So no.
• Point E: x=-3, y=-3. Solid line: 0.5*(-3) - 2 = -3.5, so y=-3 ≤ -3.5? No. Wait, this is conflicting. The key is visual: the shaded region is the gray area. Point A is in gray, so it's a solution. So the answer for Point A is "Yes, it is a solution", others (B, C, D, E, F) are "No" except A.

(Assuming the question is to check each point, here's the answer for Point A as an example, but since the problem says "which point", likely Point A is the solution.)

Final Answer (for Point A):
Yes, it is a solution