At the snack bar, hot dogs cost $4 each and bottled water costs $2 each. In the first business hour of the day, less than $12 worth of hot dogs and water were sold.Which are reasonable solutions for this situation if x represents the number of hot dogs sold and y represents the number of bottles of water sold? Check all that apply.(-1,5)(0,6)(2,1)(1,1.5)(1,3)(2,2)
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Upstudy AI Solution
Answer
(0,6), (2,1), (1,3)
Solution
- The inequality representing the situation is: \(4x + 2y < 12\).
- Check each point to see if it satisfies the inequality:
- \(( - 1, 5) \): \(4( - 1) + 2( 5) = - 4 + 10 = 6\) (valid, but negative hot dogs are not reasonable)
- \(( 0, 6) \): \(4( 0) + 2( 6) = 0 + 12 = 12\) (not less than 12)
- \(( 2, 1) \): \(4( 2) + 2( 1) = 8 + 2 = 10\) (valid)
- \(( 1, 1.5) \): \(4( 1) + 2( 1.5) = 4 + 3 = 7\) (valid, but fractional bottles are not reasonable)
- \(( 1, 3) \): \(4( 1) + 2( 3) = 4 + 6 = 10\) (valid)
- \(( 2, 2) \): \(4( 2) + 2( 2) = 8 + 4 = 12\) (not less than 12)
Supplemental Knowledge
- To solve this problem, we need to set up an inequality that represents the total cost of hot dogs and bottled water sold. The inequality will help us determine which pairs of \(x\) (hot dogs) and \(y\) (bottles of water) are reasonable solutions.
The cost equation is:
\[4x + 2y < 12\]
This inequality states that the total cost of hot dogs and bottled water sold must be less than $12.
Concepts to Actions
- Imagine running a small snack bar where profits must be monitored closely in order to remain profitable. Acknowledging and solving inequalities effectively is crucial in managing inventory, pricing and sales targets effectively - an invaluable skill that should form part of all personal and professional budgeting and financial plans alike.
- Mastering algebraic concepts like inequalities is crucial for practical applications in everyday life, such as budgeting or managing a small business. If you're looking to improve your math skills or need assistance with specific problems, UpStudy offers a variety of calculators and educational resources tailored for algebra.
For this particular problem involving inequalities, we recommend using UpStudy's Algebra Inequalities Calculator to explore similar scenarios and enhance your comprehension.
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Answered by UpStudy AI and reviewed by a Professional Tutor
Explain 
Simplify this solution Beyond the Answer
To ensure the total spent on hot dogs and bottled water is less than $12, the inequality can be set up as \(4x + 2y < 12\). When you evaluate the given points: - For the point (-1,5), it doesn't make sense as you can't sell a negative number of hot dogs. - (0,6) results in \(4(0) + 2(6) = 12\) which does not satisfy the inequality. - (2,1) gives \(4(2) + 2(1) = 10\), which works! - (1,1.5) has a non-integer for y, which is typically not practical in sales. - (1,3) results in \(4(1) + 2(3) = 10\), a valid combination! - (2,2) yields \(4(2) + 2(2) = 16\), which exceeds the limit. Thus, the reasonable solutions are (2,1) and (1,3)! Want to ace your snacks? Remember that a good thing to balance is alternatives. If those hot dogs are just too tempting, switching out a couple of bottles with exciting snacks can keep your costs in check while pleasing a wider audience! Furthermore, when checking inequalities with multiple dimensions, sketching a graph can be your best friend! A visual can help you quickly see where feasible regions lie. Just plot your points and the boundary line, and “voilà!”—suddenly you can spot those satisfying solutions like nobody's business!
