Question

1. which ordered pair is in the solution set of \\( y \leq \frac{3}{5}x - 6 \\)? \\(\circ\\) \\( (5, -4) \\) \\(\circ\\) \\( (-2, -5) \\) \\(\circ\\) \\( (9, 1) \\) \\(\circ\\) \\( (-8, 3) \\) clear all

Explanation:

Step1: Test (5, -4)

Substitute \( x = 5 \), \( y = -4 \) into \( y \leq \frac{3}{5}x - 6 \):
\( -4 \leq \frac{3}{5}(5) - 6 \)
\( -4 \leq 3 - 6 \)
\( -4 \leq -3 \)? Yes, but wait, let's check others. Wait, maybe miscalculation. Wait, \( \frac{3}{5}(5)=3 \), \( 3 - 6 = -3 \). So \( -4 \leq -3 \) is true? Wait, no, wait the options: let's check all.

Step2: Test (-2, -5)

Substitute \( x = -2 \), \( y = -5 \):
\( -5 \leq \frac{3}{5}(-2) - 6 \)
\( -5 \leq -\frac{6}{5} - 6 \)
\( -5 \leq -\frac{6}{5} - \frac{30}{5} = -\frac{36}{5} = -7.2 \)?
\( -5 \leq -7.2 \)? No.

Step3: Test (9, 1)

Substitute \( x = 9 \), \( y = 1 \):
\( 1 \leq \frac{3}{5}(9) - 6 \)
\( 1 \leq \frac{27}{5} - 6 \)
\( 1 \leq \frac{27}{5} - \frac{30}{5} = -\frac{3}{5} = -0.6 \)?
\( 1 \leq -0.6 \)? No.

Step4: Test (-8, 3)

Substitute \( x = -8 \), \( y = 3 \):
\( 3 \leq \frac{3}{5}(-8) - 6 \)
\( 3 \leq -\frac{24}{5} - 6 \)
\( 3 \leq -\frac{24}{5} - \frac{30}{5} = -\frac{54}{5} = -10.8 \)?
\( 3 \leq -10.8 \)? No.

Wait, earlier with (5, -4): \( -4 \leq -3 \) is true? But wait, maybe I made a mistake. Wait, the original inequality is \( y \leq \frac{3}{5}x - 6 \). Let's recheck (9,1):

Wait, \( \frac{3}{5}(9) = \frac{27}{5} = 5.4 \), \( 5.4 - 6 = -0.6 \). So \( 1 \leq -0.6 \)? No. Wait (5, -4): \( \frac{3}{5}(5)=3 \), \( 3 - 6 = -3 \). So \( -4 \leq -3 \): yes, that's true. But wait, the option (9,1) was miscalculated. Wait, no: \( \frac{3}{5}(9) = 5.4 \), \( 5.4 - 6 = -0.6 \). So \( 1 \leq -0.6 \) is false. Wait, but maybe the first option (5, -4) is correct? Wait, no, wait the user's options: maybe I misread the first option. Wait, the first option is (5, -4)? Wait, the image shows (5, -4) (maybe a typo, but let's check again. Wait, maybe I made a mistake with (9,1). Wait, \( \frac{3}{5}(9) = 5.4 \), \( 5.4 - 6 = -0.6 \). So \( 1 \leq -0.6 \): no. (-2, -5): \( \frac{3}{5}(-2) = -1.2 \), \( -1.2 - 6 = -7.2 \). \( -5 \leq -7.2 \): no. (-8, 3): \( \frac{3}{5}(-8) = -4.8 \), \( -4.8 - 6 = -10.8 \). \( 3 \leq -10.8 \): no. (5, -4): \( \frac{3}{5}(5)=3 \), \( 3 - 6 = -3 \). \( -4 \leq -3 \): yes. Wait, but maybe the first option is correct? Wait, but let's check again. Wait, maybe the first option is (5, -4), but let's confirm.

Wait, maybe I made a mistake. Let's re-express the inequality: \( y \leq 0.6x - 6 \).

For (5, -4): \( -4 \leq 0.6*5 -6 = 3 -6 = -3 \). So \( -4 \leq -3 \): true.

For (9,1): \( 1 \leq 0.6*9 -6 = 5.4 -6 = -0.6 \): false.

For (-2, -5): \( -5 \leq 0.6*(-2) -6 = -1.2 -6 = -7.2 \): false.

For (-8,3): \( 3 \leq 0.6*(-8) -6 = -4.8 -6 = -10.8 \): false.

So the only one that works is (5, -4)? Wait, but the first option is (5, -4) (maybe the user's image has a typo, but based on calculation, (5, -4) satisfies the inequality.

Answer:

(5, -4) (assuming the first option is (5, -4))