Inequalities in Two Variables Practice

Question

Complete this assessment to review what you’ve learned. It will not count toward your grade.
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Question
Use the image to answer the question.

The x- and y- axes of a coordinate plane are labeled ‘Adults’ and ‘Children,’ respectively, and range from 0 to 20 in increments of 5. A line connects left parenthesis 0 comma 20 right parenthesis to left parenthesis 15 comma 0 right parenthesis.

The image shows an inequality representing the possible number of adults and children in attendance at a party. Which of the labeled points represents a viable solution in this context? Enter the coordinates exactly as they appear above, in the form (x,y)
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(1 point)
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Remaining Attempts : 3
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Answer 1

To determine a viable solution in the given context, we need to assess the line connecting the points (0, 20) and (15, 0). The line represents the boundary of the inequality, and typically, solutions will lie below or on this line.

To find the coordinates of the points, you can visually test each point in the context of the inequality defined by the line:

1. Calculate the equation of the line:

   • The line has a slope of \(-\frac{20}{15} = -\frac{4}{3}\).
   • The equation can be written as \(y = -\frac{4}{3}x + 20\).

2. Choose different points (x,y) in the context of possible adults and children at the party, adhering to the non-negative constraints (both x and y should be between 0 and their respective maximums).

By evaluating potential points that fall within the area defined by the line:

• A point like (10, 5) would need to be tested for solutions where \(y \leq -\frac{4}{3}x + 20\).
• Plugging in \(x = 10\): \[ y \leq -\frac{4}{3}(10) + 20 \] \[ y \leq -\frac{40}{3} + 20 \] \[ y \leq -\frac{40}{3} + \frac{60}{3} \] \[ y \leq \frac{20}{3} \approx 6.67 \]

Since \(y = 5\) is less than \(6.67\), the point (10, 5) satisfies the inequality.

Final Answer: (10, 5)