Question

$y geq -\frac{1}{2}x$
$y < \frac{1}{2}x + 1$

\bigcirc \\ (5, -2), (3, 1), (-4, 2)
\bigcirc \\ (5, -2), (3, -1), (4, -3)
\bigcirc \\ (5, -2), (3, 1), (4, 2)
\bigcirc \\ (5, -2), (-3, 1), (4, 2)

Explanation:

Step1: Test (5,-2) in both inequalities

First inequality: $-2 \geq -\frac{1}{2}(5) \implies -2 \geq -2.5$ (True)
Second inequality: $-2 < \frac{1}{2}(5)+1 \implies -2 < 3.5$ (True)

Step2: Test (3,-1) in both inequalities

First inequality: $-1 \geq -\frac{1}{2}(3) \implies -1 \geq -1.5$ (True)
Second inequality: $-1 < \frac{1}{2}(3)+1 \implies -1 < 2.5$ (True)

Step3: Test (4,-3) in both inequalities

First inequality: $-3 \geq -\frac{1}{2}(4) \implies -3 \geq -2$ (False, skip this option)

Step4: Test (3,1) in both inequalities

First inequality: $1 \geq -\frac{1}{2}(3) \implies 1 \geq -1.5$ (True)
Second inequality: $1 < \frac{1}{2}(3)+1 \implies 1 < 2.5$ (True)
Now test (-4,2):
First inequality: $2 \geq -\frac{1}{2}(-4) \implies 2 \geq 2$ (True)
Second inequality: $2 < \frac{1}{2}(-4)+1 \implies 2 < -1$ (False)

Step5: Test (4,2) in both inequalities

First inequality: $2 \geq -\frac{1}{2}(4) \implies 2 \geq -2$ (True)
Second inequality: $2 < \frac{1}{2}(4)+1 \implies 2 < 3$ (True)
Now test (-3,1):
First inequality: $1 \geq -\frac{1}{2}(-3) \implies 1 \geq 1.5$ (False)

Step6: Verify the valid set

Only the set (5,-2), (3,-1), (4,-3) has all points satisfying both inequalities (correction: recheck (4,-3): $-3 \geq -2$ is false, recheck option 2: all points pass: (5,-2) passes, (3,-1) passes, (4,-3): wait, no—wait, recheck the second inequality for (4,-3): $-3 < \frac{1}{2}(4)+1 = 3$, which is true, but first inequality: $-3 \geq -2$ is false. Wait, recheck option 2: (3,-1) passes both, (5,-2) passes both, (4,-3) fails first. Wait, no—wait the graph: the blue line is solid ($y \geq -\frac{1}{2}x$), red is dashed ($y < \frac{1}{2}x+1$). The overlapping region is below red, above blue.

(5,-2): above blue ($-2 \geq -2.5$), below red ($-2 < 3.5$): yes.
(3,-1): above blue ($-1 \geq -1.5$), below red ($-1 < 2.5$): yes.
(4,-3): above blue ($-3 \geq -2$? No, -3 < -2, so it's below blue, not in the region. Wait, mistake: option 2's third point is (4,-3), which is below the blue line, so invalid. Wait option 3: (5,-2), (3,1), (4,2): (4,2): $2 < \frac{1}{2}(4)+1=3$, true; $2 \geq -2$, true. (3,1): $1 < 2.5$, true; $1 \geq -1.5$, true. But (3,1) is above the red line? No, $1 < 2.5$ is true. Wait no, the red line at x=3 is $y=0.5*3+1=2.5$, so (3,1) is below it, correct. But (4,2): red line at x=4 is 3, 2<3, correct. Blue line at x=4 is -2, 2≥-2, correct. But (3,1) is in the red shaded area? Wait the graph's red area is below the red line, above the blue line. (3,1) is there. But (-4,2): blue line at x=-4 is 2, so $2 \geq 2$ is true, but red line at x=-4 is $0.5*(-4)+1=-1$, 2 < -1 is false, so (-4,2) is above red line, not in region.

Wait option 2: (5,-2), (3,-1), (4,-3): (4,-3): blue line at x=4 is -2, -3 < -2, so it's below blue line, not in the region (since we need $y \geq -\frac{1}{2}x$). So that's invalid.

Wait option 3: (5,-2), (3,1), (4,2): all points satisfy both inequalities:
(5,-2): $-2 \geq -2.5$, $-2 < 3.5$: true
(3,1): $1 \geq -1.5$, $1 < 2.5$: true
(4,2): $2 \geq -2$, $2 < 3$: true

Wait I made a mistake earlier with (-4,2): option 1's third point is (-4,2), which fails the second inequality. Option 4's (-3,1): $1 \geq 1.5$ is false. So the correct set is option 3? No, wait the graph: the red shaded area is below the red line and above the blue line. (4,2) is above the blue line (yes, blue line at x=4 is y=-2) and below red line (y=3), so yes. (3,1) is below red line (y=2.5) and above blue line (y=-1.5), yes. (5,-2) is above blue line (y=-2.5) and below red line (y=3.5), yes.

Wait no, recheck option 2: (4,-3) is below blue line (y=-2), so it's in the gray area, not the red overlapping area. So option 3 is correct? Wait no, the second inequality is $y < \frac{1}{2}x+1$. For (4,2): $2 < 2+1=3$, which is true. For (3,1): $1 < 1.5+1=2.5$, true. For (5,-2): $-2 < 2.5+1=3.5$, true. All pass.

Wait but let's check the graph: the red shaded area is below the red dashed line and above the blue solid line. All three points in option 3 lie in that region.

Answer:

O (5, -2), (3, 1), (4, 2)