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) )
e ( (2, 0) )

Explanation:

Step1: Substitute A (-2, -4)

Substitute \( x = -2 \), \( y = -4 \) into \( 5x - 2y \):
\( 5(-2) - 2(-4) = -10 + 8 = -2 \).
Check if \( -2 \leq 8 \): Yes.

Step2: Substitute B (-1, -10)

Substitute \( x = -1 \), \( y = -10 \) into \( 5x - 2y \):
\( 5(-1) - 2(-10) = -5 + 20 = 15 \).
Check if \( 15 \leq 8 \): No.

Step3: Substitute C (0, -2)

Substitute \( x = 0 \), \( y = -2 \) into \( 5x - 2y \):
\( 5(0) - 2(-2) = 0 + 4 = 4 \).
Check if \( 4 \leq 8 \): Yes.

Step4: Substitute D (1, -4)

Substitute \( x = 1 \), \( y = -4 \) into \( 5x - 2y \):
\( 5(1) - 2(-4) = 5 + 8 = 13 \).
Check if \( 13 \leq 8 \): No.

Step5: Substitute E (2, 0)

Substitute \( x = 2 \), \( y = 0 \) into \( 5x - 2y \):
\( 5(2) - 2(0) = 10 - 0 = 10 \).
Wait, correction: Wait, the inequality is \( 5x - 2y \leq 8 \)? Wait, original problem says \( 5x - 2y \leq 8 \)? Wait, no, the user's problem: "inequality 5x - 2y x 8"? Wait, maybe typo, assume \( 5x - 2y \leq 8 \). Wait, but E: 10 ≤8? No. Wait, maybe the line is \( 5x - 2y = 8 \), and the inequality is \( 5x - 2y \leq 8 \) (below the line) or \( \geq \)? Wait, graph: the line \( 5x - 2y = 8 \) can be rewritten as \( y = \frac{5}{2}x - 4 \). Let's recheck substitutions:

Wait, maybe I misread the inequality. Let's re-express the line: \( 5x - 2y = 8 \implies 2y = 5x - 8 \implies y = \frac{5}{2}x - 4 \). The inequality: if the shaded region is below the line (since for \( x=0 \), \( y=-4 \), and the line goes up. Let's check each point:

A (-2, -4): Plug into \( 5x - 2y \): \( 5(-2) - 2(-4) = -10 + 8 = -2 \). Is \( -2 \leq 8 \)? Yes.

B (-1, -10): \( 5(-1) - 2(-10) = -5 + 20 = 15 \). \( 15 \leq 8 \)? No.

C (0, -2): \( 5(0) - 2(-2) = 0 + 4 = 4 \). \( 4 \leq 8 \)? Yes.

D (1, -4): \( 5(1) - 2(-4) = 5 + 8 = 13 \). \( 13 \leq 8 \)? No.

E (2, 0): \( 5(2) - 2(0) = 10 \). \( 10 \leq 8 \)? No. Wait, that can't be. Maybe the inequality is \( 5x - 2y \geq 8 \)? Let's check:

A: -2 ≥8? No.

B:15 ≥8? Yes.

C:4 ≥8? No.

D:13 ≥8? Yes.

E:10 ≥8? Yes.

Ah, likely typo: the inequality is \( 5x - 2y \geq 8 \). Let's re-express the line: \( 5x - 2y = 8 \), so the solution set for \( 5x - 2y \geq 8 \) is above the line (since the line is \( y = \frac{5}{2}x - 4 \), so \( 5x - 2y \geq 8 \implies y \leq \frac{5}{2}x - 4 \)? Wait, no: \( 5x - 2y \geq 8 \implies -2y \geq -5x + 8 \implies y \leq \frac{5}{2}x - 4 \) (dividing by -2 reverses inequality). Wait, the graph: the line has a positive slope. Let's check the points again with \( 5x - 2y \geq 8 \):

A (-2, -4): \( 5(-2) - 2(-4) = -10 + 8 = -2 \). \( -2 \geq 8 \)? No.

B (-1, -10): \( 5(-1) - 2(-10) = -5 + 20 = 15 \). \( 15 \geq 8 \)? Yes.

C (0, -2): \( 5(0) - 2(-2) = 0 + 4 = 4 \). \( 4 \geq 8 \)? No.

D (1, -4): \( 5(1) - 2(-4) = 5 + 8 = 13 \). \( 13 \geq 8 \)? Yes.

E (2, 0): \( 5(2) - 2(0) = 10 \). \( 10 \geq 8 \)? Yes.

So B, D, E satisfy \( 5x - 2y \geq 8 \). But the original problem's inequality was mistyped (probably \( \geq \) instead of \( x \)). Assuming that, the three ordered pairs are B (-1, -10), D (1, -4), E (2, 0). Wait, but let's confirm with the line equation. The line \( 5x - 2y = 8 \): when \( x=2 \), \( y=(10 - 8)/2 = 1 \)? Wait, no: \( 5(2) - 2y = 8 \implies 10 - 2y = 8 \implies 2y=2 \implies y=1 \). Wait, the graph shows a line passing through, maybe (0, -4) and (2, 1)? Wait, the graph in the image: x-axis from -10 to 10, y-axis same. The line crosses y-axis at (0, -4) (since when x=0, 5(0)-2y=8 → y=-4) and x-axis at (8/5, 0) ≈ (1.6, 0). Wait, the point E is (2, 0), which is to the right of the x-intercept. Let's recalculate E: \( 5(2) - 2(0) = 10 \), which is greater than 8, so above the line (since the line at x=2 has y=1, so (2,0) is below y=1, but the inequality: if the line is \( 5x - 2y = 8 \), then \( 5x - 2y \geq 8 \) is the region where \( y \leq \frac{5}{2}x - 4 \). At x=2, \( \frac{5}{2}(2) - 4 = 5 - 4 = 1 \). So (2,0) has y=0 ≤1, so it's in \( y \leq \frac{5}{2}x - 4 \), so \( 5x - 2y \geq 8 \) (since \( 5x - 2y \geq 8 \implies y \leq \frac{5}{2}x - 4 \)). So (2,0) is in the solution set.

D (1, -4): \( \frac{5}{2}(1) - 4 = 2.5 - 4 = -1.5 \). y=-4 ≤ -1.5? No, -4 < -1.5, so y=-4 is below -1.5, so \( 5x - 2y = 13 \geq 8 \), so yes.

B (-1, -10): \( \frac{5}{2}(-1) - 4 = -2.5 - 4 = -6.5 \). y=-10 ≤ -6.5? Yes, so \( 5x - 2y = 15 \geq 8 \), yes.

A (-2, -4): \( \frac{5}{2}(-2) - 4 = -5 - 4 = -9 \). y=-4 ≤ -9? No, -4 > -9, so \( 5x - 2y = -2 \geq 8 \)? No.

C (0, -2): \( \frac{5}{2}(0) - 4 = -4 \). y=-2 ≤ -4? No, -2 > -4, so \( 5x - 2y = 4 \geq 8 \)? No.

So the correct three are B (-1, -10), D (1, -4), E (2, 0). Wait, but earlier when I thought the inequality was \( \leq \), it was wrong. The mistyped inequality (5x - 2y x 8) is likely \( 5x - 2y \geq 8 \). So the three ordered pairs are B, D, E.

Answer:

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