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This graph shows the inequality 25x+15y≥200

and represents Lina's earnings for lawn mowing and dog walking, where x represents the number of lawns mowed and y represents the number of dogs walked. Which of the labeled points is a viable solution in this context?

(2, 3)

None of the points are a viable solution.

(20, 11.5)

(10, 10)

5 answers

To determine which of the labeled points is a viable solution for the inequality \(25x + 15y \geq 200\), we can substitute each point into the inequality and check if it satisfies the condition.

  1. Point (2, 3): \[ 25(2) + 15(3) = 50 + 45 = 95 \] \(95 \geq 200\) is false.

  2. Point (20, 11.5): \[ 25(20) + 15(11.5) = 500 + 172.5 = 672.5 \] \(672.5 \geq 200\) is true.

  3. Point (10, 10): \[ 25(10) + 15(10) = 250 + 150 = 400 \] \(400 \geq 200\) is true.

Based on this analysis, the viable solution points are (20, 11.5) and (10, 10). Therefore, both (20, 11.5) and (10, 10) are viable solutions in the context of the inequality, while (2, 3) is not.

So, if a point must be labeled and only one choice is allowed, you should indicate either (20, 11.5) or (10, 10) as viable solutions. If the question asks for just one, then you can choose either.

The correct answer to the query of which labeled point is a viable solution is:

  • (20, 11.5) or (10, 10) (either can be stated as a viable solution).

If the question states that none are viable, that is incorrect.

Which of the following points is a solution to the linear inequality 4x−y≤10
as shown on the graph?

(1 point)
Responses

(3.5, −1)

(2,−2)

(5, 0)

(3, 0)

To determine which of the points is a solution to the linear inequality \(4x - y \leq 10\), we will substitute each point into the inequality and check if it satisfies the condition.

  1. Point (3.5, -1): \[ 4(3.5) - (-1) = 14 + 1 = 15 \] \(15 \leq 10\) is false.

  2. Point (2, -2): \[ 4(2) - (-2) = 8 + 2 = 10 \] \(10 \leq 10\) is true.

  3. Point (5, 0): \[ 4(5) - 0 = 20 - 0 = 20 \] \(20 \leq 10\) is false.

  4. Point (3, 0): \[ 4(3) - 0 = 12 - 0 = 12 \] \(12 \leq 10\) is false.

Based on this analysis, the only point that satisfies the inequality \(4x - y \leq 10\) is:

  • (2, -2).

This point is a solution to the linear inequality.

Select the points that are part of the solution set for the inequality y ≤ 2/5 x + 1 Select ALL that apply.(2 points)
Responses

(10, 4)

(10, 6)

(5, 3)

(10, 5.5)

To determine which points are part of the solution set for the inequality \( y \leq \frac{2}{5}x + 1 \), we will substitute each point into the inequality and check if it satisfies the condition.

  1. Point (10, 4): \[ y = 4; \quad \text{Check: } 4 \leq \frac{2}{5}(10) + 1 \] \[ \frac{2}{5}(10) + 1 = 4 + 1 = 5 \] \[ 4 \leq 5 \quad \text{(true)} \]

  2. Point (10, 6): \[ y = 6; \quad \text{Check: } 6 \leq \frac{2}{5}(10) + 1 \] \[ \frac{2}{5}(10) + 1 = 5 \] \[ 6 \leq 5 \quad \text{(false)} \]

  3. Point (5, 3): \[ y = 3; \quad \text{Check: } 3 \leq \frac{2}{5}(5) + 1 \] \[ \frac{2}{5}(5) + 1 = 2 + 1 = 3 \] \[ 3 \leq 3 \quad \text{(true)} \]

  4. Point (10, 5.5): \[ y = 5.5; \quad \text{Check: } 5.5 \leq \frac{2}{5}(10) + 1 \] \[ \frac{2}{5}(10) + 1 = 5 \] \[ 5.5 \leq 5 \quad \text{(false)} \]

Based on this analysis, the points that are part of the solution set for the inequality \(y \leq \frac{2}{5}x + 1\) are:

  • (10, 4)
  • (5, 3)

So, the correct response includes the points (10, 4) and (5, 3).