Math Inequality: Hand Soaps And Lotions Purchase

Let's dive into a common word problem scenario that involves setting up inequalities. We're going to explore how to represent a purchase scenario using mathematical notation, specifically focusing on the number of hand soaps and lotions bought. This kind of problem is fundamental in understanding how to translate real-world situations into algebraic expressions, which is a key skill in mathematics. We'll break down the problem, analyze the given options, and figure out the best way to represent the purchase with an inequality.

Understanding the Variables and the Scenario

In this problem, we are introduced to two variables: $x$ and $y$. It's crucial to understand what each of these variables represents. $x$ is designated to stand for the number of hand soaps purchased. On the other hand, $y$ is designated to stand for the number of lotions purchased. When we combine these two quantities, we get the total number of items bought. For instance, if someone buys 5 hand soaps and 3 lotions, then $x=5$ and $y=3$. The total number of items purchased would be $x+y = 5+3 = 8$. This concept of combining variables to represent a total is a cornerstone of algebraic problem-solving.

The scenario implies a constraint or a condition related to the total number of items purchased. While the problem statement itself doesn't explicitly state the constraint (like a budget or a limit on the number of items), the multiple-choice options provide us with potential constraints. We need to determine which inequality best represents her purchase. This suggests there's an implied condition or a limit that governs the total number of hand soaps and lotions bought. The core of the problem lies in interpreting this implied condition and matching it with the correct mathematical inequality.

Think about it like this: if you go to a store and you're told you can only buy a certain number of items, or if you decide you don't want to buy too many things, you're essentially setting a limit. This limit can be a maximum number of items (you can't buy more than 12), or it could be a minimum number of items (you must buy at least 12). The wording of the options will guide us to the correct interpretation. The symbols '<', '>', '$\leq$', and '$\geq$' each have a distinct meaning in mathematics, and choosing the right one is vital for accurately modeling the situation. We'll analyze each of these symbols and how they apply to our variables $x$ and $y$ in the context of a purchase.

The foundation of solving this problem rests on correctly identifying the relationship between the total number of items ($x+y$) and the numerical value provided in the options (which is 12). Understanding this relationship is paramount to selecting the inequality that accurately reflects the situation described by the problem. We will explore each option to see how it translates into a real-world constraint and why one might be a better fit than others, assuming a standard interpretation of such problems.

Analyzing the Inequality Options

We are given four possible inequalities, each representing a different relationship between the total number of hand soaps ($x$) and lotions ($y$) and the number 12:

• A. $x+y < 12$: This inequality means that the total number of items purchased is strictly less than 12. In other words, the sum of hand soaps and lotions must be 11 or fewer. It does not include the possibility of buying exactly 12 items. For example, if someone buys 5 hand soaps and 6 lotions, $x+y=11$, which satisfies this inequality. However, if they bought 6 hand soaps and 6 lotions, $x+y=12$, this would not satisfy the inequality. This implies a strict upper limit where 12 itself is not an allowed total.

• B. $x+y > 12$: This inequality means that the total number of items purchased is strictly greater than 12. This implies that the person bought 13 or more items. For instance, buying 7 hand soaps and 7 lotions ($x+y=14$) would satisfy this inequality. However, buying 6 hand soaps and 6 lotions ($x+y=12$) would not satisfy it, nor would buying 5 hand soaps and 6 lotions ($x+y=11$). This option suggests a scenario where the purchase must exceed a certain threshold.

• C. $x+y \leq 12$: This inequality means that the total number of items purchased is less than or equal to 12. This is often referred to as "at most 12." This inequality includes the possibility of buying exactly 12 items. So, buying 5 hand soaps and 6 lotions ($x+y=11$) satisfies this. Buying 6 hand soaps and 6 lotions ($x+y=12$) also satisfies this. However, buying 7 hand soaps and 7 lotions ($x+y=14$) would not satisfy it. This represents a scenario where there is a maximum limit, and that maximum is inclusive.

• D. $x+y \geq 12$: This inequality means that the total number of items purchased is greater than or equal to 12. This is often referred to as "at least 12." This inequality includes the possibility of buying exactly 12 items. Buying 6 hand soaps and 6 lotions ($x+y=12$) satisfies this. Buying 7 hand soaps and 7 lotions ($x+y=14$) also satisfies this. However, buying 5 hand soaps and 6 lotions ($x+y=11$) would not satisfy it. This represents a scenario where there is a minimum limit, and that minimum is inclusive.

Determining the Best Representation

The phrase "which inequality best represents her purchase?" is key here. Word problems often imply common real-world constraints. Without additional context stating that she must buy a certain number or cannot buy more than a certain number, we have to infer the most typical scenario.

In many consumer scenarios, there's an upper limit to what is purchased, either due to personal choice, availability, or a special offer that might apply up to a certain quantity. For example, a store might offer a 'buy 12 or less' deal, or a customer might decide they don't want to exceed 12 items. Conversely, a scenario where someone must buy at least 12 items (meaning 11 or fewer are not acceptable) is less common unless explicitly stated (e.g., "to get a discount, you must buy at least 12 items").

Let's consider the options again in this light. Options B ($x+y > 12$) and D ($x+y \geq 12$) represent a minimum purchase requirement. This is less likely to be the default assumption for a general purchase without specific wording indicating such a requirement.

Options A ($x+y < 12$) and C ($x+y \leq 12$) represent an upper limit on the number of items. The difference lies in whether the limit of 12 itself is included. If the problem implies a maximum, and that maximum can include 12, then 'less than or equal to' is the appropriate symbol. If the maximum strictly excludes 12, then 'less than' is appropriate.

In typical mathematical word problems where a total quantity is constrained, the phrase "at most" or "no more than" translates to $\leq$, while "less than" translates to $<$. Similarly, "at least" or "no less than" translates to $\geq$, and "greater than" translates to $>$.

Since the question doesn't provide specific wording like "she bought fewer than 12 items" or "she bought at least 12 items," we must infer the most common or standard interpretation. Often, when dealing with purchasing limits, the limit itself is considered achievable. Therefore, the inequality that best represents a scenario where the total number of items bought does not exceed 12, and can be exactly 12, is $x+y \leq 12$. This is because it allows for any combination of hand soaps and lotions as long as their sum does not surpass 12, including the case where the sum is exactly 12.

This interpretation aligns with common scenarios like bulk discounts that apply up to a certain quantity, or a personal budget that restricts the total number of items to a maximum of 12. Without further context, assuming an inclusive upper bound is the most reasonable and widely applicable interpretation.

Consider a real-world example: If you're told you can get a free gift for every 12 items you buy, but you can't carry more than 12 items at once, you might buy anywhere from 0 to 12 items. This scenario is perfectly modeled by $x+y \leq 12$. The inequality simply states that the total number of items purchased is within a certain boundary, and the most common type of boundary in such contexts is an upper limit that includes the boundary value itself.

Therefore, when faced with such a problem without explicit wording, the inequality that captures the most common form of a purchase constraint is $x+y \leq 12$. It represents that the total quantity of items is 12 or less. This covers all possibilities from buying nothing ($x=0, y=0$) up to buying exactly 12 items in total, ensuring that the purchase does not exceed the implied limit.

Conclusion

We've analyzed the variables, the scenario, and the mathematical meaning of each inequality option. The key to selecting the best representation lies in understanding the typical implied constraints in purchasing scenarios. When a total number of items is considered, an upper limit that includes the boundary value is the most common and therefore the most likely interpretation in the absence of specific wording.

Thus, if $x$ represents the number of hand soaps and $y$ represents the number of lotions she bought, the inequality that best represents her purchase, implying a maximum limit of 12 items that includes 12 itself, is $x+y \leq 12$. This is option C.

For more information on inequalities and how to apply them in various contexts, you can explore resources on Khan Academy which offers excellent tutorials and practice problems on this topic.