Interpreting Inequality Solutions - Pre Algebra

1. Which scenario correctly demonstrates a valid interpretation of this solution set in a cooking context?
   Answer: Using exactly 2 cups works, 2.75 cups works, and 3.5 cups works, but 1.9 cups and 3.6 cups do not work When you encounter compound inequalities in word problems, you need to carefully interpret what the inequality symbols mean in the real-world context. The compound inequality $2 \leq f \leq 3.5$ uses "less than or equal to" symbols (≤), which means the boundary values are included in the solution set. Let's break down what $2 \leq f \leq 3.5$ tells us: the amount of flour (f) must be greater than or equal to 2 cups AND less than or equal to 3.5 cups. This means 2 cups is the minimum allowed amount, 3.5 cups is the maximum allowed amount, and any value between them also works. Choice D correctly identifies that exactly 2 cups works (satisfies f ≥ 2), 2.75 cups works (falls within the range), and exactly 3.5 cups works (satisfies f ≤ 3.5). It also correctly excludes 1.9 cups (less than 2) and 3.6 cups (greater than 3.5). Choice A incorrectly excludes the boundary values 2 and 3.5, which would only be true if the inequality used strict symbols (< and >) instead of ≤. Choice B incorrectly excludes 3.5 cups, treating the upper bound as if it were f < 3.5. Choice C incorrectly excludes exactly 2 cups, treating the lower bound as if it were f > 2. Remember: ≤ and ≥ include the boundary values, while < and > exclude them. Always check whether boundary values are part of the solution when interpreting inequalities in context.
2. If the competitor charges $23 for a delivery, which statement best describes the distances the company can deliver while staying competitive?
   Answer: The company can deliver to distances of 20 miles or less while remaining competitive with their pricing structure Setting up the inequality: $8 + 0.75d \leq 23$. Solving: $0.75d \leq 15$, so $d \leq 20$. This means the company can deliver to distances of 20 miles or less. Choice B is wrong because it's not just exactly 20 miles, but any distance up to and including 20. Choice C reverses the inequality direction. Choice D ignores that distances of 20 miles or less work.
3. If the inequality $25 + 0.10t < 40$ represents this situation and has solution $t < 150$, which statement correctly interprets what this means for the customer's texting habits?
   Answer: The customer can send any whole number of texts from 0 up to and including 149 messages per month 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.
4. A student solved the inequality $-3x + 12 > 6$ and got $x > 2$. When interpreting this solution, which statement correctly describes what values of $x$ satisfy the original inequality?
   Answer: All real numbers less than 2 satisfy the inequality, so any value like 1, 0, or -5 works perfectly The student made an error when dividing by -3. The correct solution is: $-3x + 12 > 6$, so $-3x > -6$, and dividing by -3 flips the inequality to get $x < 2$. Therefore, all real numbers less than 2 satisfy the inequality. Choice A uses the student's incorrect answer. Choice C treats it as an equation. Choice D is incorrect since there are infinitely many solutions.
5. The inequality $120 + 15w > 300$ has solution $w > 12$. If today is the start of week 1, during which week will the student first meet their savings goal?
   Answer: The student will first meet their goal during week 13, since they need more than 12 weeks When you encounter an inequality word problem, you need to distinguish between the mathematical solution and the real-world interpretation. The inequality $120 + 15w > 300$ gives us $w > 12$, meaning the student needs more than 12 weeks of saving to exceed $300. Here's the key insight: "more than 12 weeks" means any time after the 12-week mark. Since we're counting in whole weeks, the first week that satisfies $w > 12$ is week 13. Let's verify: after 12 weeks, the student has $120 + 15(12) = 300$ exactly. After 13 weeks, they have $120 + 15(13) = 315$, which exceeds their goal. Answer B is wrong because week 12 gives exactly $300, but the goal requires "more than $300." Answer A misses the point—we can determine the exact week mathematically without needing to know when they check their balance. Answer D arbitrarily adds extra weeks beyond what the problem asks for; we want the first week they meet the goal, not a week with a comfortable buffer. The correct answer is C: the student will first meet their goal during week 13, since they need more than 12 weeks. Remember this pattern: when an inequality solution gives you "greater than" a number, and you're working with discrete time periods like weeks, the first period that satisfies the inequality is the next whole number up. Don't confuse "at least 12" with "more than 12."
6. If $h$ represents the total hours parked where $h > 1$, which statement best interprets the solution $h \leq 5.8$?
   Answer: The customer can park for a maximum of 5 complete hours plus part of the 6th hour Since parking is charged by complete hours, $h \leq 5.8$ means the customer can park for at most 5 complete hours (paying $4 + 4(2.50) = 14$) plus part of the 6th hour. If they stayed a complete 6th hour, they'd pay $16.50$, exceeding their budget. Choice B incorrectly suggests 6 complete hours work. Choice C treats it as an exact requirement. Choice D ignores the budget constraint.
7. Which interpretation correctly describes how long the tank can drain before the pump stops working?
   Answer: The pump will work for any amount of time up to and including 15 hours of continuous draining Setting up: $150 - 8t \geq 30$. Solving: $-8t \geq -120$, so $t \leq 15$. This means the pump works for 15 hours or less. At exactly 15 hours, there are still 30 gallons (the minimum), so the pump still works. Choice A suggests the pump stops at exactly 15 hours. Choice C reverses the inequality. Choice D excludes the boundary case of exactly 15 hours.
8. An inequality has the solution set $x \geq -3$. A student claims this means 'x can be any number except those smaller than -3.' Which part of this interpretation needs correction?
   Answer: The phrase should specify that x includes -3 itself, not just numbers greater than -3 When working with inequalities, you need to pay careful attention to the symbols and what they include. The inequality $x \geq -3$ uses the "greater than or equal to" symbol, which has two parts: the "greater than" (>) and the "equal to" (=) components. The student's interpretation captures the "greater than" part correctly - x can indeed be any number larger than -3. However, they missed the crucial "equal to" component. The symbol $\geq$ means x can be greater than -3 OR equal to -3. So x = -3 is actually included in the solution set, along with all numbers greater than -3. Looking at the answer choices: Choice A is wrong because the interpretation is incomplete - it omits that -3 itself is included. Choice B incorrectly suggests x must be positive only, but the solution set includes negative numbers like -2, -1, and -3 itself. Choice C makes the opposite error, claiming x cannot equal -3, which would be true for $x > -3$ but not for $x \geq -3$. Choice D correctly identifies the missing piece - the interpretation should clarify that x includes -3 itself. Remember this key distinction: $\geq$ and $\leq$ include the boundary value (the number after the symbol), while $>$ and $<$ exclude it. When reading inequality solutions, always check whether the boundary point is included or excluded - this is a common source of errors in algebra problems.

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