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.

(Note: The problem mentions "THREE" ordered pairs, but only B and D satisfy? Wait, maybe a typo or missing options. Wait, re - check Option A: Wait, calculation error! Let's recalculate Option A: \(5(-2)-2(-4)=-10 + 8=-2\), correct. Wait, maybe another pair? Wait, maybe the original problem has more options, but in given options, B, D, and maybe I missed? Wait, no, the user provided A, B, C, D. Wait, maybe I made a mistake. Wait, let's re - check Option A again: \(x=-2,y = - 4\): \(5*(-2)-2*(-4)=-10 + 8=-2\), which is less than 8. Option B: \(x=-1,y=-10\): \(5*(-1)-2*(-10)=-5 + 20 = 15\geq8\). Option C: \(x = 0,y=-2\): \(5*0-2*(-2)=4\lt8\). Option D: \(x = 1,y=-4\): \(5*1-2*(-4)=5 + 8 = 13\geq8\). Wait, maybe there is a mistake in the problem or my calculation. Wait, maybe the line is \(5x-2y = 8\), and the inequality is \(5x - 2y\geq8\), so the region is above the line (since for the line \(5x-2y = 8\), solving for \(y\): \(y=\frac{5x - 8}{2}\), so the inequality \(5x - 2y\geq8\) is \(y\leq\frac{5x - 8}{2}\)). Let's check the y - value for each x:

For Option A: \(x=-2\), \(y=\frac{5*(-2)-8}{2}=\frac{-10 - 8}{2}=-9\). The point (-2, -4): \(y=-4\), and \(-4\gt - 9\), so it is above the line, so \(5x - 2y=-2\lt8\), so not in solution.

For Option B: \(x=-1\), \(y=\frac{5*(-1)-8}{2}=\frac{-5 - 8}{2}=-6.5\). The point (-1, -10): \(y=-10\lt - 6.5\), so below the line, so \(5x - 2y = 15\geq8\), in solution.

For Option C: \(x = 0\), \(y=\frac{5*0-8}{2}=-4\). The point (0, -2): \(y=-2\gt - 4\), above the line, \(5x - 2y = 4\lt8\), not in solution.

For Option D: \(x = 1\), \(y=\frac{5*1-8}{2}=\frac{5 - 8}{2}=-1.5\). The point (1, -4): \(y=-4\lt - 1.5\), below the line, \(5x - 2y = 13\geq8\), in solution.

Wait, maybe there is a missing option, but among the given, B and D satisfy. But the problem says "THREE". Maybe a typo. But according to the calculation, B, D, and maybe another? Wait, maybe I miscalculated Option A. No, \(5*(-2)-2*(-4)=-10 + 8=-2\). Wait, maybe the original problem has more options, but in the given options, B and D are correct. But the user's problem says "THREE", so maybe there is an error. But based on the given options, B, D, and maybe I missed? Wait, no, the user provided A, B, C, D. So maybe the answer is B, D, and maybe another? Wait, no, let's check again. Wait, maybe Option A: \(x=-2,y=-4\): \(5*(-2)-2*(-4)=-10 + 8=-2\), which is less than 8. Option B: 15≥8, yes. Option C: 4 < 8, no. Option D:13≥8, yes. So two options. But the problem says three. Maybe a mistake in the problem. But according to the calculation, B and D are correct. If we assume that there is a mistake and maybe Option A is correct, but no, calculation shows otherwise. Wait, maybe the inequality is \(5x - 2y\leq8\)? No, the problem says \(5x - 2y\geq8\). So based on the given options, B and D are in the solution set. But the problem says three, so maybe there is a missing option. But with the given options, the correct ones are B and D. But since the problem says three, maybe I made a mistake. Wait, let's check Option A again: \(x=-2,y=-4\): \(5*(-2)-2*(-4)=-10 + 8=-2\), which is less than 8. Option B:15≥8, yes. Option C:4 < 8, no. Option D:13≥8, yes. So two options. Maybe the original problem has a typo. But according to the calculation, B and D are correct.

Answer:

B. (-1, -10), D. (1, -4) (Note: There seems to be an issue as only two of the given options satisfy the inequality, but based on calculation, these are the correct ones among the provided options.)