Question

1. which points are solutions to the inequality $y < -\frac{1}{2}x + 4$ ? (2 points)

circle all possible answers
(4, 2) (-3, 6)
(-2, -3) (1, -5)
graph of a line on a coordinate plane with grid, points plotted on the line and a grid below for plotting or checking points

Explanation:

Step1: Test point (4,2)

Substitute \(x = 4\), \(y = 2\) into \(y<-\frac{1}{2}x + 4\):
\(2<-\frac{1}{2}(4)+4\)
\(2<-2 + 4\)
\(2<2\)? No, wait, \(2 = 2\), but the inequality is \(y<-\frac{1}{2}x + 4\) (strictly less than). Wait, maybe miscalculation. Wait, \(-\frac{1}{2}(4)=-2\), \(-2 + 4 = 2\). So \(2<2\) is false? But the original has (4,2) circled. Wait, maybe the inequality is \(y\leq-\frac{1}{2}x + 4\)? Or maybe I made a mistake. Wait, let's check other points.

Step2: Test point (-3,6)

Substitute \(x=-3\), \(y = 6\):
\(6<-\frac{1}{2}(-3)+4\)
\(6<\frac{3}{2}+4\)
\(6<\frac{11}{2}=5.5\)? No, \(6>5.5\), so not a solution.

Step3: Test point (-2,-3)

Substitute \(x=-2\), \(y=-3\):
\(-3<-\frac{1}{2}(-2)+4\)
\(-3<1 + 4\)
\(-3<5\): True. So (-2,-3) is a solution.

Step4: Test point (1,-5)

Substitute \(x = 1\), \(y=-5\):
\(-5<-\frac{1}{2}(1)+4\)
\(-5<-\frac{1}{2}+4\)
\(-5<\frac{7}{2}=3.5\): True. So (1,-5) is a solution.

Wait, the original (4,2): \(2<-\frac{1}{2}(4)+4=2\), so \(2<2\) is false. Maybe the inequality was \(y\leq-\frac{1}{2}x + 4\)? If so, (4,2) would be equal. But based on strict inequality \(y<-\frac{1}{2}x + 4\), let's re-express:

Wait, maybe the graph: the line is \(y = -\frac{1}{2}x + 4\), dashed or solid? The graph shows solid dots, so maybe \(y\leq-\frac{1}{2}x + 4\). Let's recheck (4,2): \(2\leq2\), true. Then (-2,-3): \(-3\leq5\), true. (1,-5): \(-5\leq3.5\), true. (-3,6): \(6\leq-\frac{1}{2}(-3)+4=5.5\)? No, 6>5.5. So if inequality is \(y\leq-\frac{1}{2}x + 4\), solutions are (4,2), (-2,-3), (1,-5). But the problem says \(y<-\frac{1}{2}x + 4\) (strict). Let's check again:

For (4,2): \(y = 2\), right-hand side (RHS) = \( -\frac{1}{2}(4)+4 = -2 + 4 = 2\). So \(2<2\) is false.

For (-2,-3): RHS = \( -\frac{1}{2}(-2)+4 = 1 + 4 = 5\). \(-3<5\): true.

For (1,-5): RHS = \( -\frac{1}{2}(1)+4 = -0.5 + 4 = 3.5\). \(-5<3.5\): true.

For (-3,6): RHS = \( -\frac{1}{2}(-3)+4 = 1.5 + 4 = 5.5\). \(6<5.5\): false.

So if strict inequality, solutions are (-2,-3), (1,-5). If non-strict, (4,2), (-2,-3), (1,-5). But the original problem's graph has solid dots, so maybe \(y\leq-\frac{1}{2}x + 4\). Let's assume the problem might have a typo or the inequality is non-strict. But based on the given options and the graph (solid line), let's re-evaluate:

Wait, the user's image: the line is solid, so inequality is \(y\leq-\frac{1}{2}x + 4\). So:

- (4,2): \(2\leq2\): true.
- (-3,6): \(6\leq5.5\): false.
- (-2,-3): \(-3\leq5\): true.
- (1,-5): \(-5\leq3.5\): true.

But the original has (4,2) circled, (-3,6) crossed, (-2,-3) crossed? Wait, the image shows (-2,-3) crossed, (1,-5) crossed? Wait, maybe I misread the points. Let's look again:

Points: (4,2) circled, (-3,6) crossed, (-2,-3) crossed, (1,-5) crossed? Wait, no, the handwriting: (4,2) is circled, (-3,6) has an X, (-2,-3) has an X, (1,-5) has an X? But that contradicts. Wait, maybe the user made a mistake, but let's solve based on the inequality \(y<-\frac{1}{2}x + 4\).

Wait, maybe I miscalculated (4,2):

\(y = 2\), \(x = 4\):

\(2 < -\frac{1}{2}(4) + 4\)

\(2 < -2 + 4\)

\(2 < 2\): No, equal. So not a solution.

(-2,-3):

\(-3 < -\frac{1}{2}(-2) + 4\)

\(-3 < 1 + 4\)

\(-3 < 5\): Yes.

(1,-5):

\(-5 < -\frac{1}{2}(1) + 4\)

\(-5 < 3.5\): Yes.

(-3,6):

\(6 < -\frac{1}{2}(-3) + 4\)

\(6 < 1.5 + 4\)

\(6 < 5.5\): No.

So the solutions are (-2, -3) and (1, -5). But the original (4,2) is circled, maybe the inequality was \(y \leq -\frac{1}{2}x + 4\), in which case (4,2) is a solution (since \(2 \leq 2\)). Let's confirm with the graph: the line is solid, so the inequality includes equality. So if \(y \leq -\frac{1}{2}x + 4\), then:

(4,2): \(2 \leq 2\): yes.

(-2,-3): \(-3 \leq 5\): yes.

(1,-5): \(-5 \leq 3.5\): yes.

(-3,6): \(6 \leq 5.5\): no.

So depending on the inequality (strict or non-strict), but the graph has solid dots, so non-strict. But the problem says \(y < -\frac{1}{2}x + 4\). Maybe a mistake in the problem, but let's proceed with the strict inequality as given.

Wait, maybe the user's graph: the point (4,2) is on the line, so if the inequality is \(y < -\frac{1}{2}x + 4\), it's not a solution. But if \(y \leq\), it is. Let's check the other points again.

Alternative approach: For a point to be a solution to \(y < -\frac{1}{2}x + 4\), it must lie below the line \(y = -\frac{1}{2}x + 4\).

- (4,2): on the line (since \(2 = -\frac{1}{2}(4) + 4\)), so not below.
- (-3,6): above the line (6 > 5.5), so not a solution.
- (-2,-3): below the line (-3 < 5), so solution.
- (1,-5): below the line (-5 < 3.5), so solution.

So the solutions are (-2, -3) and (1, -5). But the original (4,2) is circled, maybe the inequality was \(y \leq\). Assuming the problem might have a typo, but based on the given inequality \(y < -\frac{1}{2}x + 4\), the solutions are (-2, -3) and (1, -5).

Answer:

The points that are solutions to \(y < -\frac{1}{2}x + 4\) are \((-2, -3)\) and \((1, -5)\). If the inequality were \(y \leq -\frac{1}{2}x + 4\), \((4, 2)\) would also be a solution. (Based on strict inequality as given, the solutions are \((-2, -3)\) and \((1, -5)\).)