practice interpreting two - variable linear inequalities.\ny < 2x + 3\nwhich of the points are solutions to…

practice interpreting two - variable linear inequalities.\ny < 2x + 3\nwhich of the points are solutions to the inequality?\ncheck all that apply.\n□ (-3, 3)\n□ (-2, -2)\n□ (-1, 1)\n□ (0, 1)\n□ (2, 5)

practice interpreting two - variable linear inequalities.\ny < 2x + 3\nwhich of the points are solutions to the inequality?\ncheck all that apply.\n□ (-3, 3)\n□ (-2, -2)\n□ (-1, 1)\n□ (0, 1)\n□ (2, 5)

Answer

Explanation:

Step1: Check point (-3, 3)

Substitute (x = -3), (y = 3) into (y < 2x + 3):
(3 < 2(-3) + 3)
(3 < -6 + 3)
(3 < -3) → False.

Step2: Check point (-2, -2)

Substitute (x = -2), (y = -2) into (y < 2x + 3):
(-2 < 2(-2) + 3)
(-2 < -4 + 3)
(-2 < -1) → True.

Step3: Check point (-1, 1)

Substitute (x = -1), (y = 1) into (y < 2x + 3):
(1 < 2(-1) + 3)
(1 < -2 + 3)
(1 < 1) → False (since inequality is strict (<)).

Step4: Check point (0, 1)

Substitute (x = 0), (y = 1) into (y < 2x + 3):
(1 < 2(0) + 3)
(1 < 3) → True.

Step5: Check point (2, 5)

Substitute (x = 2), (y = 5) into (y < 2x + 3):
(5 < 2(2) + 3)
(5 < 4 + 3)
(5 < 7) → True? Wait, wait—wait, (2(2)+3=7), (5 < 7) is true? Wait, no, wait the graph: the line is dashed, and the shaded region? Wait, the inequality is (y < 2x + 3). Wait, (x=2), (y=5): (5 < 4 + 3) → (5 < 7), which is true. Wait, but let's recheck. Wait, maybe I made a mistake earlier. Wait, no—wait the graph: the line is (y = 2x + 3), dashed, and the shaded region? Wait, the original problem: the inequality is (y < 2x + 3). Let's recheck each point:

Wait, (-2, -2): (y=-2), (2x+3=2(-2)+3=-1), so (-2 < -1) → true.

(0,1): (y=1), (2x+3=3), (1 < 3) → true.

(2,5): (y=5), (2x+3=7), (5 < 7) → true? Wait, but let's check the graph. The line is dashed, and the shaded region—wait, maybe the shaded region is above? No, the inequality is (y < 2x + 3), so below the line. Wait, the line (y=2x+3) has slope 2, y-intercept 3. At (x=2), (y=7), so the point (2,5) is below (y=7), so it should satisfy (y < 2x + 3). Wait, but let's re-express:

Wait, maybe I messed up the first point. Let's recheck (-3,3): (2(-3)+3=-3), so (3 < -3) → false. Correct.

(-1,1): (2(-1)+3=1), so (1 < 1) → false (since it's strict inequality). Correct.

(-2, -2): (2(-2)+3=-1), (-2 < -1) → true. Correct.

(0,1): (2(0)+3=3), (1 < 3) → true. Correct.

(2,5): (2(2)+3=7), (5 < 7) → true. Wait, but the graph: the shaded region—wait, maybe the shaded region is above? No, the inequality is (y < 2x + 3), so below the line. Wait, the line is dashed, so the region is (y < 2x + 3) (below). So (2,5) is below (y=7), so it's in the region. Wait, but let's confirm with the inequality:

Yes, (y < 2x + 3) means all points where (y) is less than (2x + 3). So:

(-3,3): 3 < -3? No.

(-2,-2): -2 < -1? Yes.

(-1,1): 1 < 1? No (strict inequality).

(0,1): 1 < 3? Yes.

(2,5): 5 < 7? Yes.

Wait, but maybe the graph's shaded region is different? Wait, the problem's graph: the line is dashed, and the shaded region is to the right/above? Wait, no, the inequality is (y < 2x + 3), so the region below the line. Let's plot the line: when (x=0), (y=3); (x=-1), (y=1); (x=-2), (y=-1); (x=2), (y=7). So the point (-2, -2) is below (y=-1) (since (-2 < -1)), so in the region. (0,1) is below (y=3) (1 < 3), so in the region. (2,5) is below (y=7) (5 < 7), so in the region. Wait, but maybe the original problem's graph has a different shaded region? Wait, the user's image: the inequality is (y < 2x + 3), and the graph shows a dashed line with shaded region. Wait, maybe I made a mistake with (2,5). Wait, let's recalculate (2x + 3) when (x=2): (2*2 + 3 = 7), so (y=5) is less than 7, so (5 < 7) is true. So (2,5) should be a solution. Wait, but let's check again.

Wait, maybe the problem's options: let's re-express each point:

  1. (-3, 3): (y=3), (2x+3=-3) → 3 < -3? No.

  2. (-2, -2): (y=-2), (2x+3=-1) → -2 < -1? Yes.

  3. (-1, 1): (y=1), (2x+3=1) → 1 < 1? No (strict inequality).

  4. (0, 1): (y=1), (2x+3=3) → 1 < 3? Yes.

  5. (2, 5): (y=5), (2x+3=7) → 5 < 7? Yes.

Wait, but maybe the graph's shaded region is above? No, the inequality is (y < 2x + 3), so below. Wait, maybe the user's graph has a typo, but according to the inequality, these points should satisfy. Wait, but let's check the initial calculation again. Wait, (2,5): (5 < 2*2 + 3) → (5 < 7), which is true. So the solutions are (-2, -2), (0, 1), (2, 5)? Wait, but let's check the graph again. The line is (y = 2x + 3), dashed, and the shaded region—if the shaded region is above the line, then the inequality would be (y > 2x + 3), but the problem says (y < 2x + 3). Wait, maybe the graph is mislabeled, but according to the inequality given ((y < 2x + 3)), we have to go by the inequality.

Wait, maybe I made a mistake with (2,5). Let's check the value of (2x + 3) at (x=2): (2*2 + 3 = 7), so (y=5) is less than 7, so it satisfies (y < 2x + 3). So the correct points are (-2, -2), (0, 1), (2, 5)? Wait, but let's check the first point again. Wait, (-2, -2): (x=-2), (y=-2). (2x + 3 = -4 + 3 = -1). So (-2 < -1) → true. (0,1): (1 < 3) → true. (2,5): (5 < 7) → true. (-3,3): (3 < -3) → false. (-1,1): (1 < 1) → false. So the solutions are (-2, -2), (0, 1), (2, 5). Wait, but maybe the graph's shaded region is different. Wait, the line is dashed, so the inequality is strict. Let's confirm with the inequality:

(y < 2x + 3)

So for each point (x, y), check if (y) is less than (2x + 3).

  • (-3, 3): 3 < -3? No.
  • (-2, -2): -2 < -1? Yes.
  • (-1, 1): 1 < 1? No.
  • (0, 1): 1 < 3? Yes.
  • (2, 5): 5 < 7? Yes.

Answer:

B. (-2, -2), D. (0, 1), E. (2, 5)
(Note: Assuming the options are labeled as A: (-3,3), B: (-2,-2), C: (-1,1), D: (0,1), E: (2,5))