Inequality For Glass Length A Mathematical Explanation

In mathematics, inequalities are used to express the relative order of two values or expressions. Unlike equations, which assert that two quantities are exactly equal, inequalities indicate that one quantity is greater than, less than, greater than or equal to, or less than or equal to another. Inequalities play a crucial role in various fields, including algebra, calculus, and real-world applications where constraints and ranges are essential.

When dealing with physical dimensions, such as the length of a piece of glass intended to fit within a frame, inequalities become particularly relevant. Consider a scenario where the length, denoted as $x$, of a piece of glass must adhere to specific constraints to fit correctly within an existing frame. The problem states that the glass length must be longer than 12 cm but not longer than 12.2 cm. This condition implies two distinct constraints: the glass length must exceed 12 cm, and it must be less than or equal to 12.2 cm. To represent these constraints mathematically, we utilize inequality symbols.

The phrase "longer than 12 cm" translates to the inequality $x > 12$, indicating that the value of $x$ is strictly greater than 12. Conversely, the phrase "not longer than 12.2 cm" implies that the glass length can be equal to 12.2 cm or any value less than it. This is represented by the inequality $x \le 12.2$, where the symbol $\le$ signifies "less than or equal to." Combining these two constraints, we arrive at a compound inequality that accurately captures the acceptable range for the glass length. A compound inequality combines two or more inequalities using the words "and" or "or." In this context, the glass length must satisfy both conditions simultaneously: it must be greater than 12 cm and less than or equal to 12.2 cm. This is mathematically expressed as $12 < x \le 12.2$. This inequality indicates that $x$ lies between 12 and 12.2, excluding 12 but including 12.2. Therefore, any glass length within this range will fit appropriately within the frame, adhering to the specified constraints.

To effectively tackle mathematical problems, especially those involving inequalities, a systematic approach is essential. When faced with a word problem, it is crucial to first dissect the information provided, identify the key variables, and translate the verbal statements into mathematical expressions. This process involves a careful examination of the language used, particularly phrases that indicate relationships such as greater than, less than, at least, or at most. In the given problem, the core task is to determine the appropriate inequality that represents the acceptable lengths of a piece of glass intended to fit within a frame. The problem explicitly states that the length, denoted as $x$, must be longer than 12 cm but not longer than 12.2 cm. These phrases form the foundation for constructing the inequality. Breaking down these statements is the first critical step in solving the problem.

The phrase "longer than 12 cm" directly translates to a mathematical inequality. The term "longer than" implies that the length $x$ must be greater than 12. Mathematically, this is represented as $x > 12$. This inequality establishes a lower bound for the acceptable glass length; it must exceed 12 cm to fit the frame properly. On the other hand, the phrase "not longer than 12.2 cm" introduces an upper bound for the glass length. The term "not longer than" signifies that the length can be equal to 12.2 cm or any value less than it. This is represented by the inequality $x \le 12.2$. This inequality includes 12.2 cm as a possible length, along with all lengths below it, ensuring that the glass does not exceed the maximum allowable dimension for the frame. By carefully dissecting these phrases and converting them into mathematical inequalities, we create a clear and precise representation of the problem's constraints. This initial step of translating verbal statements into mathematical expressions is crucial for setting up the problem and proceeding towards a solution. With these individual inequalities established, the next step involves combining them to form a compound inequality that accurately captures the entire range of acceptable glass lengths.

Constructing the inequality involves combining the individual inequalities derived from the problem statement. As established earlier, the length $x$ of the glass must satisfy two conditions simultaneously: it must be greater than 12 cm ($x > 12$), and it must be less than or equal to 12.2 cm ($x \le 12.2$). To represent these conditions collectively, we use a compound inequality. Compound inequalities combine two or more inequalities using the words "and" or "or." In this case, the word "and" is appropriate because the glass length must satisfy both conditions concurrently. It must be greater than 12 cm and less than or equal to 12.2 cm.

To construct the compound inequality, we combine the two individual inequalities using the appropriate symbols and connect them logically. The inequality $x > 12$ indicates that $x$ is greater than 12, while the inequality $x \le 12.2$ indicates that $x$ is less than or equal to 12.2. Combining these two conditions, we obtain the compound inequality $12 < x \le 12.2$. This inequality concisely expresses the acceptable range for the glass length. The notation $12 < x$ signifies that $x$ is strictly greater than 12, excluding 12 itself. The notation $x \le 12.2$ signifies that $x$ is less than or equal to 12.2, including 12.2 as a possible value. Therefore, the compound inequality $12 < x \le 12.2$ reads as "$x$ is greater than 12 and less than or equal to 12.2." This means that any glass length within this range will fit the frame, adhering to the specified constraints. This inequality provides a clear and concise representation of the allowable lengths, making it easier to determine whether a particular piece of glass will fit or not. The construction of this compound inequality is a crucial step in solving the problem, as it translates the verbal constraints into a mathematical form that can be readily understood and applied.

To fully understand the solution, it is essential to interpret the meaning of the inequality $12 < x \le 12.2$ in the context of the problem. This inequality represents the range of acceptable lengths for the piece of glass to fit within the frame. Understanding the solution involves recognizing the boundaries and the inclusiveness or exclusiveness of those boundaries. The inequality $12 < x \le 12.2$ indicates that the length $x$ must be greater than 12 cm but can be at most 12.2 cm. This means that the glass length cannot be exactly 12 cm, as the inequality symbol is a strict "greater than" ($>$). However, the length can be equal to 12.2 cm, as the inequality symbol includes "equal to" ($\le$).

The lower bound of the acceptable range is 12 cm, but this value is not included in the solution set. This is because the inequality $x > 12$ specifies that the length must be strictly greater than 12 cm. If the glass were exactly 12 cm long, it would not satisfy the condition of being "longer than 12 cm." Therefore, any length slightly above 12 cm, such as 12.01 cm, 12.1 cm, or 12.15 cm, would be acceptable. The upper bound of the acceptable range is 12.2 cm, and this value is included in the solution set. The inequality $x \le 12.2$ allows for the length to be equal to 12.2 cm. This means that a piece of glass that is exactly 12.2 cm long would fit within the frame, as it satisfies the condition of being "not longer than 12.2 cm." Any length less than 12.2 cm but greater than 12 cm would also be acceptable. For example, lengths such as 12.18 cm, 12.19 cm, or 12.2 cm would all fit within the frame.

In practical terms, the inequality $12 < x \le 12.2$ defines a narrow range of acceptable lengths for the glass. This range allows for slight variations in the glass length while ensuring that it fits securely within the frame. A glass cutter would need to cut the glass precisely within this range to ensure a proper fit. This understanding of the solution not only provides the mathematical answer but also offers a practical interpretation of the constraints, highlighting the importance of precision and accuracy in real-world applications. The inequality serves as a guide for determining whether a given length is suitable, emphasizing the practical implications of mathematical concepts in everyday scenarios.

Inequalities, such as the one derived for the glass length, have numerous real-world applications beyond simple mathematical exercises. Real-world applications of inequalities span various fields, including engineering, economics, and everyday problem-solving. The ability to represent constraints and ranges using inequalities is crucial for decision-making and problem-solving in diverse scenarios. In engineering, inequalities are used to define tolerance limits for manufactured parts. For example, the diameter of a bolt may need to fall within a specific range to ensure it fits properly in a machine. The acceptable range can be expressed as an inequality, setting both upper and lower bounds for the diameter. If the bolt's diameter falls outside this range, it may not function correctly, leading to potential mechanical failures.

In economics, inequalities are used to model budget constraints and resource allocation. For instance, a consumer may have a limited budget to spend on goods and services. This budget constraint can be represented as an inequality, where the total expenditure must be less than or equal to the available budget. This helps individuals and businesses make informed decisions about spending and saving. Inequalities are also used in optimization problems, where the goal is to maximize or minimize a certain quantity subject to constraints. For example, a company may want to maximize its profit while adhering to constraints on production capacity, labor costs, and material availability. These constraints can be expressed as inequalities, and the optimization problem can be solved using mathematical techniques such as linear programming.

In everyday problem-solving, inequalities are used to make decisions involving time, distance, and quantity. For example, a person planning a trip may have a limited amount of time to reach their destination. The distance to the destination and the speed of travel can be used to calculate the travel time, which must be less than or equal to the available time. This constraint can be represented as an inequality, helping the person plan their route and schedule accordingly. Similarly, inequalities can be used to determine the amount of ingredients needed for a recipe, the number of items to purchase within a budget, or the maximum weight that can be carried in a vehicle. These examples illustrate the widespread applicability of inequalities in real-world situations. The ability to formulate and solve inequalities is a valuable skill that enhances problem-solving capabilities across various domains, making it an essential tool for both professionals and everyday individuals.

In conclusion, the problem of determining the appropriate inequality for the glass length demonstrates the practical application of mathematical concepts in real-world scenarios. By carefully breaking down the problem statement and translating the verbal constraints into mathematical expressions, we were able to construct the compound inequality $12 < x \le 12.2$. This inequality accurately represents the acceptable range of lengths for the glass to fit within the frame, highlighting the importance of precision and attention to detail in practical applications. Understanding the solution involves not only deriving the inequality but also interpreting its meaning in the context of the problem. The inequality specifies that the glass length must be greater than 12 cm but can be at most 12.2 cm, excluding 12 cm but including 12.2 cm. This understanding allows for informed decision-making and ensures that the glass is cut to the appropriate dimensions for a proper fit.

The broader implications of inequalities extend beyond this specific problem. Inequalities are fundamental tools in various fields, including engineering, economics, and everyday problem-solving. They enable the representation of constraints, the definition of acceptable ranges, and the optimization of solutions subject to limitations. From setting tolerance limits in manufacturing to modeling budget constraints in economics, inequalities play a crucial role in decision-making and problem-solving across diverse domains. The ability to formulate and solve inequalities is a valuable skill that enhances critical thinking and analytical abilities, empowering individuals to tackle real-world challenges effectively. Therefore, a thorough understanding of inequalities and their applications is essential for both academic pursuits and practical endeavors.