Inequalities in Two Variables Quick Check

5 of 55 of 5 Items

Question
Use the image to answer the question.

The x- and y- axes of a coordinate plane range from 0 to 30 and 0 to 20 in increments of 5 and are labeled 'Lawns' and 'Dogs,' respectively. A line connects left parenthesis 0 comma 10 right parenthesis and left parenthesis 8 comma 0 right parenthesis.

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?

(1 point)
Responses

(20,11.5)
left parenthesis 20 comma 11.5 right parenthesis

None of the points are a viable solution.
None of the points are a viable solution.

(10,10)
left parenthesis 10 comma 10 right parenthesis

(2,3)
left parenthesis 2 comma 3 right parenthesis
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1 answer

To determine which of the labeled points is a viable solution to the inequality \(25x + 15y \geq 200\), we need to evaluate each point by substituting the coordinates into the inequality.

  1. For the point (20, 11.5): \[ 25(20) + 15(11.5) = 500 + 172.5 = 672.5 \] Since \(672.5 \geq 200\), this point is a viable solution.

  2. For the point (10, 10): \[ 25(10) + 15(10) = 250 + 150 = 400 \] Since \(400 \geq 200\), this point is also a viable solution.

  3. For the point (2, 3): \[ 25(2) + 15(3) = 50 + 45 = 95 \] Since \(95 < 200\), this point is not a viable solution.

Based on the evaluation, both (20, 11.5) and (10, 10) are viable solutions. Since the question asks for one viable solution and both are valid:

The best answer from the given options would be:

  • (20, 11.5) or (10, 10) depending on whether you need to select one or both. If you must choose one, either is valid based on the context given.

However, if only one option needs to be provided from the response choices, you can select either (20, 11.5) or (10, 10) as they both satisfy the inequality. If the choice of "None of the points are a viable solution" is also included, this option would be incorrect since viable solutions exist.

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