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Kiani is shopping at a bookstore with a budget of $60. Hardbacks cost $12 and paperbacks cost $5. Write a two-variable inequality and determine the constraints of the problem. Which of the following points represents a viable solution?(1 point) Responses (1,4.5) left parenthesis 1 comma 4.5 right parenthesis (6,6) left parenthesis 6 comma 6 right parenthesis (−2,6) left parenthesis negative 2 comma 6 right parenthesis (2,4)

3 answers

Let \( x \) represent the number of hardbacks Kiani buys, and \( y \) represent the number of paperbacks. The cost of the books can be expressed with the following inequality based on Kiani's budget:

\[ 12x + 5y \leq 60 \]

Constraints of the Problem

  1. Non-negativity constraint: Kiani cannot buy a negative number of books. Therefore, we have:

    • \( x \geq 0 \)
    • \( y \geq 0 \)

  2. Budget constraint: The total cost of hardbacks and paperbacks must not exceed her budget of $60:

    • \( 12x + 5y \leq 60 \)

Testing Potential Solutions

Now we'll evaluate each of the given points to see if they satisfy the inequality \( 12x + 5y \leq 60 \).

  1. Point (1, 4.5): \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \leq 60 \quad \text{(true)} \]

  2. Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \leq 60 \quad \text{(false)} \]

  3. Point (−2, 6): \[ 12(-2) + 5(6) = -24 + 30 = 6 \leq 60 \quad \text{(true, but not viable due to negative x)} \]

  4. Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \leq 60 \quad \text{(true)} \]

Conclusion

The points (1, 4.5), (−2, 6), and (2, 4) satisfy the budget constraint, but only (1, 4.5) and (2, 4) are viable because they do not involve buying negative quantities of books. So, the viable solutions given the constraints are (1, 4.5) and (2, 4).

Final Viable Solution

Among the choices, (1, 4.5) and (2, 4) are viable solutions. If you need to choose just one from the options, either (1, 4.5) or (2, 4) would be correct.

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 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 (20,11.5) left parenthesis 20 comma 11.5 right parenthesis (2,3)

To determine which of the labeled points is a viable solution for the inequality \( 25x + 15y \geq 200 \), we'll test each of the points given.

  1. Point (10, 10): \[ 25(10) + 15(10) = 250 + 150 = 400 \geq 200 \quad \text{(true)} \]

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

  3. Point (2, 3): \[ 25(2) + 15(3) = 50 + 45 = 95 \geq 200 \quad \text{(false)} \]

Conclusion

The points (10, 10) and (20, 11.5) both satisfy the inequality. Therefore, these two points are viable solutions.

Since the prompt asks which labeled point is a viable solution, either of the first two options \( (10, 10) \) or \( (20, 11.5) \) would be correct answers. The statement "None of the points are a viable solution" is incorrect.