Question

the graph of the line $5x - 2y = 8$ is shown. which three ordered pairs are in the solution set of the inequality $5x - 2y \geq 8$?\
a $(-2, -4)$\
b $(-1, -10)$\
c $(0, -2)$\
d $(1, -4)$

Explanation:

Step1: Test Option A (-2, -4)

Substitute \( x = -2 \), \( y = -4 \) into \( 5x - 2y \):
\( 5(-2) - 2(-4) = -10 + 8 = -2 \).
Check if \( -2 \geq 8 \)? No.

Step2: Test Option B (-1, -10)

Substitute \( x = -1 \), \( y = -10 \) into \( 5x - 2y \):
\( 5(-1) - 2(-10) = -5 + 20 = 15 \).
Check if \( 15 \geq 8 \)? Yes.

Step3: Test Option C (0, -2)

Substitute \( x = 0 \), \( y = -2 \) into \( 5x - 2y \):
\( 5(0) - 2(-2) = 0 + 4 = 4 \).
Check if \( 4 \geq 8 \)? No.

Step4: Test Option D (1, -4)

Substitute \( x = 1 \), \( y = -4 \) into \( 5x - 2y \):
\( 5(1) - 2(-4) = 5 + 8 = 13 \).
Check if \( 13 \geq 8 \)? Yes.

Wait, there's a mistake—maybe I missed a point or miscalculated. Let's re - check the line \( 5x - 2y = 8 \), or maybe the inequality's solution region. The line \( 5x - 2y = 8 \) can be rewritten as \( y=\frac{5}{2}x - 4 \). The inequality \( 5x - 2y\geq8 \) is \( y\leq\frac{5}{2}x - 4 \) (after rearranging: \( - 2y\geq - 5x + 8\), then \( y\leq\frac{5}{2}x - 4 \)).

Let's re - test each point:

Option A (-2, -4):

\( y=-4 \), \( \frac{5}{2}(-2)-4=-5 - 4=-9 \). Is \( -4\leq - 9 \)? No.

Option B (-1, -10):

\( y = - 10 \), \( \frac{5}{2}(-1)-4=-\frac{5}{2}-4=-\frac{13}{2}=-6.5 \). Is \( -10\leq - 6.5 \)? Yes.

Option C (0, -2):

\( y=-2 \), \( \frac{5}{2}(0)-4=-4 \). Is \( -2\leq - 4 \)? No.

Option D (1, -4):

\( y = - 4 \), \( \frac{5}{2}(1)-4=\frac{5}{2}-4=-\frac{3}{2}=-1.5 \). Is \( -4\leq - 1.5 \)? Yes.

Wait, but the problem says "THREE" ordered pairs. Maybe there's a typo or I missed an option. Wait, maybe the original problem has more options? Wait, the given options are A, B, C, D. Wait, maybe I made a mistake in the first calculation for Option A. Wait, \( 5x - 2y\geq8 \). For A: \( 5(-2)-2(-4)=-10 + 8=-2 \), \( -2\geq8 \)? No. For B: 15≥8: Yes. For C: 4≥8: No. For D:13≥8: Yes. Wait, maybe there's another option. Wait, maybe the line is \( 5x-2y = 8 \), and the solution region is above or below? Wait, when we have \( 5x - 2y\geq8 \), let's take a test point not on the line, say (0,0): \( 0 - 0=0\geq8 \)? No. So the solution region is where \( 5x - 2y\geq8 \), which is the region that satisfies the inequality. Wait, maybe the options are mis - presented, or I made a mistake. Wait, let's check the value of \( 5x - 2y \) again for each point:

- A: \( 5*(-2)-2*(-4)=-10 + 8=-2 \). \( -2\geq8 \)? No.
- B: \( 5*(-1)-2*(-10)=-5 + 20 = 15 \). \( 15\geq8 \)? Yes.
- C: \( 5*0-2*(-2)=0 + 4 = 4 \). \( 4\geq8 \)? No.
- D: \( 5*1-2*(-4)=5 + 8 = 13 \). \( 13\geq8 \)? Yes.

Wait, maybe there's a mistake in the problem, or maybe I missed an option. But according to the given options, only B and D satisfy. But the problem says "THREE". Wait, maybe the original line is \( 5x + 2y = 8 \)? No, the problem says \( 5x - 2y = 8 \). Wait, maybe the points are different. Wait, let's check the graph. The line \( 5x - 2y = 8 \) has a y - intercept at \( y=-4 \) (when \( x = 0 \), \( -2y = 8\), \( y=-4 \)) and x - intercept at \( x=\frac{8}{5}=1.6 \). The inequality \( 5x - 2y\geq8 \) is the region above or below? Let's solve for y: \( -2y\geq - 5x + 8\), \( y\leq\frac{5}{2}x - 4 \). So the region below the line. Let's take a point on the line: (0, - 4). \( 5(0)-2(-4)=8 \), which satisfies \( 8\geq8 \). Oh! Wait, I missed that the line is included (since it's \( \geq \)). So for the point (0, - 4), but that's not an option. Wait, Option C is (0, - 2). Wait, no. Wait, let's recalculate Option C: \( 5(0)-2(-2)=0 + 4 = 4 \), \( 4\geq8 \)? No. Wait, Option A: (-2, -4): \( 5(-2)-2(-4)=-10 + 8=-2 \), \( -2\geq8 \)? No. Option B: (-1, -10): 15≥8: Yes. Option D: (1, -4):13≥8: Yes. Wait, maybe the problem has a typo, or I made a mistake. Alternatively, maybe the inequality is \( 5x - 2y\leq8 \)? No, the problem says \( \geq \).

Wait, maybe the original options are different. Wait, perhaps the user made a mistake in pasting. But based on the given options, only B and D satisfy. But the problem says "THREE". Maybe there's an error. However, assuming that there's a mistake in my calculation for Option A or C. Wait, let's check Option A again: \( x=-2 \), \( y = - 4 \). \( 5x-2y=5*(-2)-2*(-4)=-10 + 8=-2 \). \( -2\geq8 \)? No. Option C: \( x = 0 \), \( y=-2 \). \( 5*0-2*(-2)=4 \). \( 4\geq8 \)? No. Option B: 15≥8: Yes. Option D:13≥8: Yes. So maybe the problem has a mistake, or maybe I missed an option. But according to the given options, the two that satisfy are B and D. But the problem says "THREE". Maybe the original line is \( 5x + 2y = 8 \)? Let's check: For \( 5x + 2y\geq8 \). Option A: \( 5(-2)+2(-4)=-10 - 8=-18 \), \( -18\geq8 \)? No. Option B: \( 5(-1)+2(-10)=-5 - 20=-25 \), \( -25\geq8 \)? No. Option C: \( 5(0)+2(-2)=-4 \), \( -4\geq8 \)? No. Option D: \( 5(1)+2(-4)=5 - 8=-3 \), \( -3\geq8 \)? No. So that's not it.

Wait, going back, the line is \( 5x - 2y = 8 \), so the solution to \( 5x - 2y\geq8 \) includes the line and the region where \( 5x - 2y\) is greater than or equal to 8. Let's find another point. Let's take (2,1): \( 5*2-2*1 = 10 - 2 = 8 \), which satisfies \( 8\geq8 \). But that's not an option.

Given the options, the two that satisfy are B (-1, -10) and D (1, -4). But the problem says "THREE". Maybe there's a mistake in the problem or in my analysis. However, based on the calculations:

- B: \( 5(-1)-2(-10)=15\geq8 \): Yes.
- D: \( 5(1)-2(-4)=13\geq8 \): Yes.
- Wait, maybe I made a mistake with Option A. Let's re - express the inequality: \( 5x - 2y\geq8 \) is equivalent to \( 5x\geq2y + 8 \). For A: \( 5(-2)=-10 \), \( 2(-4)+8=-8 + 8 = 0 \). Is \( -10\geq0 \)? No. For B: \( 5(-1)=-5 \), \( 2(-10)+8=-20 + 8=-12 \). Is \( -5\geq - 12 \)? Yes (since - 5 is greater than - 12). For D: \( 5(1)=5 \), \( 2(-4)+8=-8 + 8 = 0 \). Is \( 5\geq0 \)? Yes. For C: \( 5(0)=0 \), \( 2(-2)+8=-4 + 8 = 4 \). Is \( 0\geq4 \)? No.

Ah! Wait a minute, I was comparing to 8, but when we rearrange \( 5x - 2y\geq8 \), it's \( 5x\geq2y + 8 \). So for each point, we can check if \( 5x\geq2y + 8 \).

- Option A: \( 5(-2)=-10 \), \( 2(-4)+8=-8 + 8 = 0 \). \( -10\geq0 \)? No.
- Option B: \( 5(-1)=-5 \), \( 2(-10)+8=-20 + 8=-12 \). \( -5\geq - 12 \)? Yes (because - 5 is to the right of - 12 on the number line, so it's greater).
- Option C: \( 5(0)=0 \), \( 2(-2)+8=-4 + 8 = 4 \). \( 0\geq4 \)? No.
- Option D: \( 5(1)=5 \), \( 2(-4)+8=-8 + 8 = 0 \). \( 5\geq0 \)? Yes.

But also, the point on the line (x,y) where \( 5x - 2y = 8 \) satisfies the inequality. Let's find a point on the line. When \( x = 2 \), \( 5(2)-2y=8\Rightarrow10 - 2y=8\Rightarrow2y = 2\Rightarrow y = 1 \). So (2,1) is on the line and satisfies \( 8\geq8 \). But it's not an option.

Wait, maybe the original problem has a different set of options. Given the current options, the two that satisfy are B and D. But the problem says "THREE". There must be an error. However, if we assume that the inequality is \( 5x - 2y\leq8 \), then:

- Option A: - 2≤8: Yes.
- Option B:15≤8: No.
- Option C:4≤8: Yes.
- Option D:13≤8: No.

But the problem says \( \geq \).

Alternatively, maybe I misread the options. Let's check the options again:

A. (-2, -4)
B. (-1, -10)
C. (0, -2)
D. (1, -4)

Wait, let's calculate \( 5x - 2y \) for each:

- A: \( 5*(-2)-2*(-4)=-10 + 8=-2 \). \( -2\geq8 \)? No.
- B: \( 5*(-1)-2*(-10)=-5 + 20 = 15 \). \( 15\geq8 \)? Yes.
- C: \( 5*0-2*(-2)=0 + 4 = 4 \). \( 4\geq8 \)? No.
- D: \( 5*1-2*(-4)=5 + 8 = 13 \). \( 13\geq8 \)? Yes.

Wait, maybe there's a third option. Wait, what about (2,1): \( 5*2-2*1 = 8 \), which satisfies \( 8\geq8 \), but it's not an option. Alternatively, maybe the line is \( 5x + 2y = 8 \). Let's check:

- A: \( 5*(-2)+2*(-4)=-10 - 8=-18 \leq8 \): Yes.
- B: \( 5*(-1)+2*(-10)=-5 - 20=-25 \leq8 \): Yes.
- C: \( 5*0+2*(-2)=-4 \leq8 \): Yes.
- D: \( 5*1+2*(-4)=5 - 8=-3 \leq8 \): Yes.

But that's not the case.

Given the problem as stated, the two ordered pairs that satisfy \( 5x - 2y\geq8 \) are B (-1, -10) and D (1, -4). But the problem says "THREE", so there must be an error. However, if we consider that the line is included and we made a mistake in Option C:

Wait, Option C is (0, -2). \( 5*0-2*(-2)=4 \). \( 4\geq8 \)? No. Option A: - 2≥8? No.

Alternatively, maybe the inequality is \( 5x - 2y\geq - 8 \). Then:

- A: - 2≥ - 8: Yes.
- B:15≥ - 8: Yes.
- C:4≥ - 8: Yes.
- D:13≥ - 8: Yes.

But the problem says \( \geq8 \).

Given the information, the correct ordered pairs among the given options are B (-1, -10) and D (1, -4). But since the problem asks for three, there might be a mistake. However, based on the calculations, the two that satisfy are B and D. But if we assume that there's a typo and the inequality is \( 5x - 2y\leq8 \), then A, C, and maybe B or D. But the problem says \( \geq \).

Answer:

B. (-1, -10), D. (1, -4) (Note: There seems to be an error in the problem as only two of the given options satisfy the inequality. If we consider the line is included and there's a miscalculation, but based on the given options, these two are correct.)