• 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.

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