Solving -5 < -3h + 7 ≤ 22 A Step-by-Step Guide To Compound Inequalities
Introduction
In mathematics, compound inequalities combine two or more inequalities into a single statement. Understanding how to solve these inequalities is crucial for various mathematical applications. This article delves into solving the compound inequality -5 < -3h + 7 ≤ 22, providing a comprehensive, step-by-step approach to ensure clarity and understanding. We will break down the process, explain the underlying principles, and offer insights into interpreting the solution. Compound inequalities, which often appear in algebra and calculus, are mathematical expressions that combine two or more inequalities. Mastering these inequalities is essential for anyone looking to excel in mathematics. This guide aims to provide you with a thorough understanding of how to solve them, using the specific example of -5 < -3h + 7 ≤ 22. Solving compound inequalities may seem challenging at first, but with a systematic approach, it becomes quite manageable. This article will equip you with the necessary tools and knowledge to tackle such problems confidently. Understanding the solution to the compound inequality -5 < -3h + 7 ≤ 22 not only helps in mathematical problem-solving but also enhances analytical skills applicable in various fields. We aim to make the process as straightforward as possible, offering clear explanations and practical steps. This introduction sets the stage for a detailed exploration of solving compound inequalities, focusing on the example at hand while also providing broader insights into the topic.
Understanding Compound Inequalities
A compound inequality is a mathematical statement that combines two or more inequalities. These inequalities are typically linked by the words "and" or "or." In the case of "and," the solution must satisfy both inequalities simultaneously, while for "or," the solution must satisfy at least one of the inequalities. Our inequality, -5 < -3h + 7 ≤ 22, is a compound inequality joined by an implied "and," meaning we are looking for values of h that satisfy both -5 < -3h + 7 and -3h + 7 ≤ 22. Before diving into the solution, it’s crucial to understand the structure of compound inequalities. They represent a range of values rather than a single value, which is the case in simple equations. The inequality -5 < -3h + 7 ≤ 22 essentially sandwiches the expression -3h + 7 between two bounds, -5 and 22. This type of representation is common in many areas of mathematics and science, where constraints and limits are important considerations. Recognizing this structure is the first step in effectively solving such problems. The ability to interpret compound inequalities correctly is fundamental. They describe intervals on the number line, and the solution to a compound inequality is often an interval or a union of intervals. Grasping this concept allows for a more intuitive approach to solving these inequalities, moving beyond mere algebraic manipulation to a deeper understanding of what the inequality represents. Compound inequalities can appear in various forms, but the basic principle remains the same: we are seeking the values that make the entire statement true. This section serves as a foundation for the subsequent steps, ensuring that the reader has a solid grasp of the nature of compound inequalities before we proceed with the solution.
Step 1: Isolate the Variable Term
To solve the compound inequality -5 < -3h + 7 ≤ 22, the first step is to isolate the term containing the variable h. This involves performing the same operations on all parts of the inequality to maintain its balance. In this case, we start by subtracting 7 from all three parts: -5 - 7 < -3h + 7 - 7 ≤ 22 - 7. This simplifies to -12 < -3h ≤ 15. Isolating the variable term is a fundamental technique in solving inequalities, much like in solving equations. The key is to perform the same operation across the entire inequality to ensure that the relationship between the expressions remains valid. Subtracting 7 from all parts of the inequality is a direct application of this principle. It effectively moves us closer to isolating the term with h. This step is crucial because it simplifies the inequality, making it easier to work with. By removing the constant term (+7) from the middle, we can focus on the variable term -3h. This isolation is a critical step in the overall solution process. The result, -12 < -3h ≤ 15, sets the stage for the next operation, which will involve dealing with the coefficient of h. Understanding this isolation process is essential not just for this particular problem, but for solving any inequality where terms need to be rearranged to isolate the variable. It’s a core skill in algebra and a building block for more advanced mathematical concepts.
Step 2: Divide by the Coefficient of the Variable (and Flip the Inequality Signs)
The next step in solving -12 < -3h ≤ 15 is to divide all parts of the inequality by the coefficient of h, which is -3. However, it is crucial to remember that dividing (or multiplying) an inequality by a negative number reverses the direction of the inequality signs. Thus, we have: -12 / -3 > -3h / -3 ≥ 15 / -3. This simplifies to 4 > h ≥ -5. The most important aspect of this step is understanding why the inequality signs flip when dividing by a negative number. This is because multiplying or dividing by a negative number reflects the number line, changing the order of the numbers. Therefore, to maintain the truth of the inequality, the direction of the signs must be reversed. Failing to do so will result in an incorrect solution. The operation itself is straightforward: divide each part of the inequality by -3. However, the conceptual understanding of sign reversal is critical. The result, 4 > h ≥ -5, is a valid solution but is often rewritten in a more conventional form for clarity. This rewriting will be addressed in the next step. Dividing by the coefficient of the variable is a common step in solving inequalities, but the rule about flipping the signs when dividing by a negative number is a potential pitfall. Mastering this concept is crucial for accurate problem-solving in inequalities. This step highlights the importance of attention to detail and understanding the underlying mathematical principles.
Step 3: Rewrite the Inequality in Standard Form
While 4 > h ≥ -5 is a correct solution, it is typically rewritten in standard form to make it easier to understand and interpret. The standard form usually presents the variable increasing from left to right. Therefore, we rewrite 4 > h ≥ -5 as -5 ≤ h < 4. This form clearly shows that h is greater than or equal to -5 and less than 4. The standard form of writing inequalities is more than just a matter of convention; it aids in clear communication and understanding of the solution. Presenting the variable in ascending order (-5 ≤ h < 4) makes it immediately clear the range of values that h can take. This is particularly useful when graphing the solution on a number line or using the solution in further calculations. Rewriting the inequality also helps avoid common misinterpretations. The original form (4 > h ≥ -5) might lead some to think of two separate inequalities rather than a single continuous range. The standard form clarifies that we are dealing with a single interval where h is bounded by -5 and 4. This step underscores the importance of presentation in mathematics. A well-presented solution is not only easier to understand but also less prone to errors in subsequent steps. The transformation from 4 > h ≥ -5 to -5 ≤ h < 4 is a simple yet crucial step in ensuring clarity and accuracy. In summary, rewriting the inequality in standard form is a best practice that enhances the interpretability and usability of the solution.
Step 4: Interpret the Solution
The solution to the compound inequality -5 < -3h + 7 ≤ 22 is -5 ≤ h < 4. This means that h can be any real number between -5 (inclusive) and 4 (exclusive). On a number line, this would be represented by a closed circle at -5 (indicating inclusion) and an open circle at 4 (indicating exclusion), with a line connecting the two. Interpreting the solution is as important as finding it. The inequality -5 ≤ h < 4 describes a range of values, not just a single value. This range includes all numbers from -5 up to, but not including, 4. The inclusion of -5 is denoted by the "≤" sign, while the exclusion of 4 is denoted by the "<" sign. Visualizing this solution on a number line can provide a clearer understanding. The closed circle at -5 signifies that -5 itself is part of the solution, while the open circle at 4 indicates that 4 is not included. The line connecting these two points represents all the numbers in between, which are also solutions to the inequality. This interpretation is crucial for applying the solution in practical contexts. For example, if h represents a physical quantity, knowing the range of possible values can be critical for making decisions or predictions. The solution set is not just a mathematical abstraction; it represents real possibilities and constraints. This step emphasizes the importance of connecting mathematical solutions to their real-world implications. The ability to interpret solutions correctly is a key skill in mathematics and its applications. In conclusion, understanding that -5 ≤ h < 4 means h can be any number in the specified range, including -5 but not 4, is essential for a complete understanding of the solution.
Conclusion
In conclusion, solving the compound inequality -5 < -3h + 7 ≤ 22 involves a series of systematic steps: isolating the variable term, dividing by the coefficient (remembering to flip the inequality signs if the coefficient is negative), and rewriting the inequality in standard form. The solution, -5 ≤ h < 4, represents all real numbers between -5 (inclusive) and 4 (exclusive). This process not only provides the solution to this specific problem but also illustrates the general method for solving compound inequalities. Mastering these steps is crucial for success in algebra and related fields. The ability to confidently solve compound inequalities is a valuable skill that extends beyond the classroom. These types of problems appear in various contexts, from engineering to economics, where understanding ranges and constraints is essential. The step-by-step approach outlined in this article provides a clear and reliable method for tackling these inequalities. Each step, from isolating the variable term to interpreting the solution, plays a critical role in the overall process. The solution itself is more than just a number; it represents a range of possibilities. Understanding how to interpret this range is key to applying the solution effectively. The inequality -5 ≤ h < 4 tells us not only the bounds of h but also which values are included and excluded. This level of detail is essential for many applications. In summary, solving compound inequalities is a fundamental skill in mathematics, and the ability to approach these problems systematically leads to accurate and meaningful solutions. This article has provided a comprehensive guide to solving the specific inequality -5 < -3h + 7 ≤ 22, while also offering broader insights into the general principles of solving compound inequalities. By following these steps and understanding the underlying concepts, you can confidently tackle similar problems in the future.
