Question

for each ordered pair (x, y), determine whether it is a solution to the inequality 8x - 2y ≤ -12.
(x, y) | is it a solution?
| yes | no
(0, 7) | ○ | ○
(-2, 4) | ○ | ○
(1, -5) | ○ | ○
(-3, -6) | ○ | ○

Explanation:

Step1: For (0, 7)

Substitute \( x = 0 \), \( y = 7 \) into \( 8x - 2y \leq -12 \).
Calculate \( 8(0) - 2(7) = 0 - 14 = -14 \).
Check if \( -14 \leq -12 \): Yes, so (0,7) is a solution? Wait, no: Wait, -14 is less than or equal to -12? Wait, -14 is less than -12, so yes? Wait, no: Wait, 8x -2y = -14. Is -14 ≤ -12? Yes, because -14 is less than -12. Wait, but let's recheck. Wait, 80 -27 = -14. -14 ≤ -12: yes, because -14 is less than -12. Wait, but maybe I made a mistake. Wait, the inequality is 8x -2y ≤ -12. So -14 ≤ -12: yes. So (0,7) is a solution? Wait, no, wait: -14 is less than -12, so yes. Wait, but let's check the next one.

Step2: For (-2, 4)

Substitute \( x = -2 \), \( y = 4 \) into \( 8x - 2y \).
Calculate \( 8(-2) - 2(4) = -16 - 8 = -24 \).
Check if \( -24 \leq -12 \): Yes, because -24 is less than -12.

Step3: For (1, -5)

Substitute \( x = 1 \), \( y = -5 \) into \( 8x - 2y \).
Calculate \( 8(1) - 2(-5) = 8 + 10 = 18 \).
Check if \( 18 \leq -12 \): No, because 18 is greater than -12.

Step4: For (-3, -6)

Substitute \( x = -3 \), \( y = -6 \) into \( 8x - 2y \).
Calculate \( 8(-3) - 2(-6) = -24 + 12 = -12 \).
Check if \( -12 \leq -12 \): Yes, because -12 is equal to -12.

Wait, but earlier for (0,7): 80 -27 = -14. -14 ≤ -12: yes. Wait, but let's confirm again.

Wait, (0,7): 8x -2y = -14. -14 ≤ -12: true. So (0,7) is Yes? Wait, but maybe I messed up. Wait, let's re-express the inequality: 8x -2y ≤ -12. Let's solve for y: -2y ≤ -12 -8x → y ≥ 6 +4x (dividing by -2 reverses the inequality). So for (0,7): y =7, 6 +4*0=6. 7 ≥6: yes. So that's correct. So (0,7) is a solution.

For (-2,4): y=4, 6 +4*(-2)=6-8=-2. 4 ≥ -2: yes. So 8x -2y = -24 ≤ -12: yes.

For (1,-5): y=-5, 6 +4*1=10. -5 ≥10? No. So 8x -2y=18 ≤-12? No.

For (-3,-6): y=-6, 6 +4*(-3)=6-12=-6. -6 ≥-6: yes. So 8x -2y=-12 ≤-12: yes.

So the answers are:

(0,7): Yes

(-2,4): Yes

(1,-5): No

(-3,-6): Yes

But let's check the calculations again:

(0,7): 80 -27 = -14. -14 ≤ -12: yes.

(-2,4): 8(-2) -24 = -16 -8 = -24. -24 ≤ -12: yes.

(1,-5): 81 -2(-5) =8 +10=18. 18 ≤-12: no.

(-3,-6):8(-3) -2(-6)= -24 +12= -12. -12 ≤-12: yes.

So the table should be:

(x,y) | Yes | No

(0,7) | ○ (Yes) | ○

(-2,4) | ○ (Yes) | ○

(1,-5) | ○ | ○ (No)

(-3,-6) | ○ (Yes) | ○

Answer:

For (0, 7): Yes
For (-2, 4): Yes
For (1, -5): No
For (-3, -6): Yes

(To mark in the table:

• (0, 7): Select "Yes"
• (-2, 4): Select "Yes"
• (1, -5): Select "No"
• (-3, -6): Select "Yes" )