Prealgebra Question of the Day

1. The customer must send fewer than 149 messages per month to stay within their budget constraints
2. The customer can send any whole number of texts from 0 up to and including 150 messages per month
3. The customer can send any whole number of texts from 0 up to and including 149 messages per month (correct answer)
4. The customer must send exactly 149 messages per month to maximize their plan's value efficiently

Explanation: When interpreting inequality solutions in real-world contexts, you need to carefully consider what the mathematical solution means in practical terms, especially when dealing with discrete quantities like text messages. The inequality $25 + 0.10t < 40$ gives us $t < 150$, meaning the number of text messages must be less than 150. Since you can only send whole numbers of text messages, this translates to 0, 1, 2, 3, ... up to 149 messages. The key insight is that 150 itself is not included because the inequality is strictly less than, not less than or equal to. Choice C correctly captures this: you can send any whole number of texts from 0 up to and including 149 messages per month, which keeps your total bill under $40. Choice A incorrectly suggests you must send fewer than 149 messages, which would exclude 149 as an option when it's actually acceptable. Choice B makes the critical error of including 150 messages, which would result in a bill of exactly $40, violating the "less than$40" requirement. Choice D misinterprets the problem entirely by suggesting you must send exactly 149 messages, when the inequality allows for any number from 0 to 149. Remember that when inequalities involve real-world quantities that must be whole numbers, always check whether the boundary value satisfies the original constraint. Strict inequalities (< or >) exclude the boundary value, while inclusive inequalities (≤ or ≥) include it.