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: Check (0,7)

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

Step2: Check (-2,4)

Substitute \( x = -2 \), \( y = 4 \) into \( 8x - 2y \).
\( 8(-2) - 2(4) = -16 - 8 = -24 \).
Check \( -24 \leq -12 \): Yes, \( -24 < -12 \).

Step3: Check (1, -5)

Substitute \( x = 1 \), \( y = -5 \) into \( 8x - 2y \).
\( 8(1) - 2(-5) = 8 + 10 = 18 \).
Check \( 18 \leq -12 \): No, 18 > -12.

Step4: Check (-3, -6)

Substitute \( x = -3 \), \( y = -6 \) into \( 8x - 2y \).
\( 8(-3) - 2(-6) = -24 + 12 = -12 \).
Check \( -12 \leq -12 \): Yes, \( -12 = -12 \).

Wait, wait, earlier for (0,7): \( 8(0) - 2(7) = -14 \). Is \( -14 \leq -12 \)? Yes, because \( -14 < -12 \). So (0,7) is a solution? Wait, but let's confirm:

For (0,7): \( 8(0) - 2(7) = -14 \). \( -14 \leq -12 \): True. So "Yes".

For (-2,4): \( 8(-2) - 2(4) = -16 - 8 = -24 \). \( -24 \leq -12 \): True. "Yes".

For (1, -5): \( 8(1) - 2(-5) = 8 + 10 = 18 \). \( 18 \leq -12 \): False. "No".

For (-3, -6): \( 8(-3) - 2(-6) = -24 + 12 = -12 \). \( -12 \leq -12 \): True. "Yes".

Wait, but maybe I messed up (0,7). Let's recheck: \( 8x - 2y \leq -12 \). \( x=0 \), \( y=7 \): \( 0 - 14 = -14 \). Is \( -14 \leq -12 \)? Yes, because \( -14 \) is less than \( -12 \), so the inequality holds. So (0,7) is "Yes"? Wait, but let's check with the inequality. The inequality is \( 8x - 2y \leq -12 \). So if the left-hand side is less than or equal to -12, it's a solution. So:

• (0,7): \( -14 \leq -12 \): Yes.
• (-2,4): \( -24 \leq -12 \): Yes.
• (1, -5): \( 18 \leq -12 \): No.
• (-3, -6): \( -12 \leq -12 \): Yes.

Wait, but maybe I made a mistake with (0,7). Let's see: \( -14 \) is less than \( -12 \), so \( -14 \leq -12 \) is true. So (0,7) is a solution. But let's check the arithmetic again. \( 8*0 = 0 \), \( 2*7 = 14 \), so \( 0 - 14 = -14 \). Correct. \( -14 \leq -12 \): Yes.

Wait, but maybe the original problem's inequality is \( 8x - 2y \leq -12 \), so let's confirm each:

1. (0,7): \( 8(0) - 2(7) = -14 \). \( -14 \leq -12 \): True (Yes).
2. (-2,4): \( 8(-2) - 2(4) = -16 - 8 = -24 \). \( -24 \leq -12 \): True (Yes).
3. (1, -5): \( 8(1) - 2(-5) = 8 + 10 = 18 \). \( 18 \leq -12 \): False (No).
4. (-3, -6): \( 8(-3) - 2(-6) = -24 + 12 = -12 \). \( -12 \leq -12 \): True (Yes).

So the answers are:

(0,7): Yes

(-2,4): Yes

(1, -5): No

(-3, -6): Yes

But let's check again for (0,7). Maybe I misread the inequality. Is it \( 8x - 2y \leq -12 \) or \( 8x - 2y \geq -12 \)? No, the problem says \( 8x - 2y \leq -12 \). So \( -14 \leq -12 \) is true. So (0,7) is Yes.

Wait, but maybe the user made a typo, but according to the problem, it's \( 8x - 2y \leq -12 \). So proceeding with that.

Answer:

For each ordered pair:

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

(To mark in the table: For each row, select "Yes" or "No" as above.)