Solving The Compound Inequality 5 < (1/2)a - 7 < A

This article delves into the process of solving the compound inequality 5 < (1/2)a - 7 < a. Compound inequalities, which combine two or more inequalities, require a systematic approach to isolate the variable and determine the solution set. In this particular case, we're dealing with a double inequality, where a central expression, (1/2)a - 7, is bounded by two values, 5 and a. We will explore the step-by-step methods to solve this inequality and express the solution in various forms, including inequality notation and interval notation. Solving inequalities is a fundamental concept in algebra, with applications across various mathematical disciplines and real-world scenarios. Understanding how to manipulate inequalities, isolate variables, and interpret solution sets is crucial for problem-solving in mathematics, physics, economics, and other fields. In this article, we will not only focus on the mechanical steps of solving the inequality but also emphasize the underlying principles and reasoning behind each step. By understanding the logic behind the operations, you'll be better equipped to tackle similar problems and apply these concepts in more complex situations. We will also discuss common pitfalls and mistakes to avoid when solving inequalities, such as the importance of reversing the inequality sign when multiplying or dividing by a negative number. Furthermore, we will illustrate how the solution set can be represented graphically on a number line, providing a visual understanding of the range of values that satisfy the inequality. By the end of this article, you should have a solid grasp of how to solve compound inequalities like 5 < (1/2)a - 7 < a and be able to confidently apply these skills to other mathematical problems.

Step 1: Isolate the Term with 'a' in the Middle

To begin solving the compound inequality 5 < (1/2)a - 7 < a, our initial goal is to isolate the term containing the variable 'a' in the middle section of the inequality. This involves performing algebraic operations on all parts of the inequality to eliminate the constant term, which in this case is -7. The key principle here is to maintain the balance of the inequality; whatever operation we perform on one part, we must perform on all parts. This ensures that the inequality remains valid throughout the solution process. To eliminate the -7, we add 7 to all three parts of the inequality: the left side (5), the middle part ((1/2)a - 7), and the right side (a). This gives us: 5 + 7 < (1/2)a - 7 + 7 < a + 7. Simplifying this, we get 12 < (1/2)a < a + 7. Now, the term with 'a' is partially isolated in the middle. We still need to deal with the coefficient (1/2) in front of 'a'. Before we address the coefficient, it's important to understand why we perform the same operation on all parts of the inequality. Inequalities represent a range of values, not just a single value. By adding the same number to all parts, we're essentially shifting the entire range without changing the relative relationships between the values. This is analogous to shifting a number line; the distances between numbers remain the same. In contrast, if we only added 7 to the middle part, we would be changing the fundamental relationship expressed by the inequality. The left and right sides would no longer accurately represent the bounds of the middle expression. Therefore, consistency in operations is crucial for preserving the integrity of the inequality and arriving at a correct solution. This step of isolating the variable term is a common starting point for solving various types of inequalities, including linear, quadratic, and compound inequalities. Mastering this technique lays a solid foundation for tackling more complex problems in algebra and beyond. In the next step, we will focus on eliminating the coefficient of 'a' to further isolate the variable and move closer to the final solution.

Step 2: Eliminate the Fraction

Having isolated the term with 'a' to some extent, our inequality now reads 12 < (1/2)a < a + 7. The next step in solving this compound inequality involves eliminating the fraction (1/2) that is multiplying 'a'. This is crucial for simplifying the inequality and making it easier to isolate 'a' completely. To eliminate the fraction, we multiply all parts of the inequality by the denominator, which in this case is 2. Remember, as with the previous step, it is essential to perform the same operation on all parts of the inequality to maintain its balance and validity. Multiplying each part by 2, we get: 2 * 12 < 2 * (1/2)a < 2 * (a + 7). Simplifying this expression yields: 24 < a < 2a + 14. Now, the fraction has been successfully eliminated, and we have a simpler inequality to work with. The inequality now involves 'a' in two parts, which means we'll need to separate it into two simpler inequalities to solve for 'a'. Before we proceed, let's discuss why multiplying by a positive number doesn't change the direction of the inequality signs. Inequalities express the relative order of values; for instance, 5 < 7 means that 5 is less than 7. When we multiply both sides of an inequality by a positive number, we are essentially scaling the values, but their relative order remains the same. If we multiply both 5 and 7 by 2, we get 10 and 14, and the relationship 10 < 14 still holds true. However, it's crucial to remember that multiplying or dividing by a negative number does change the direction of the inequality sign. This is because multiplying by a negative number reverses the order of the values on the number line. For example, if we multiply 5 < 7 by -1, we get -5 > -7. This is a critical rule to remember when solving inequalities. In our current case, we multiplied by a positive number (2), so the inequality signs remain unchanged. The next step will involve separating the compound inequality into two simpler inequalities and solving each one individually. This will allow us to determine the range of values for 'a' that satisfy the original inequality.

Step 3: Separate the Compound Inequality

At this stage, we have simplified the inequality to 24 < a < 2a + 14. This compound inequality essentially combines two separate inequalities: 24 < a and a < 2a + 14. To solve for 'a', we need to treat these as two distinct inequalities and solve each one independently. This separation is a crucial step in handling compound inequalities, as it allows us to isolate the variable and determine its possible range of values more effectively. Let's first consider the inequality 24 < a. This inequality is already in a relatively simple form; it directly states that 'a' must be greater than 24. There's no further algebraic manipulation needed here. However, it's often helpful to rewrite it as a > 24 to make it clearer that we're looking for values of 'a' that are larger than 24. Now, let's turn our attention to the second inequality: a < 2a + 14. To solve this, we need to isolate 'a' on one side of the inequality. We can start by subtracting 'a' from both sides: a - a < 2a + 14 - a, which simplifies to 0 < a + 14. Next, we subtract 14 from both sides to isolate 'a': 0 - 14 < a + 14 - 14, resulting in -14 < a. Again, it's helpful to rewrite this as a > -14 for clarity. Now we have two inequalities: a > 24 and a > -14. To find the solution to the original compound inequality, we need to find the values of 'a' that satisfy both of these conditions simultaneously. This is because the original inequality 24 < a < 2a + 14 requires 'a' to be both greater than 24 and satisfy the condition a < 2a + 14. Graphically, this means we're looking for the intersection of the solution sets of the two individual inequalities on the number line. Before we determine the intersection, let's recap why we separate compound inequalities in this way. Compound inequalities with 'and' (as is the case here, since 'a' must satisfy both conditions) represent the overlap between the solutions of the individual inequalities. Separating them allows us to solve each condition independently and then identify the common solution set. In the next step, we will combine the solutions of the two inequalities to find the overall solution to the original compound inequality.

Step 4: Combine the Solutions

After separating the compound inequality, we arrived at two individual inequalities: a > 24 and a > -14. Now, the crucial step is to combine these solutions to find the values of 'a' that satisfy both inequalities simultaneously. This is because the original compound inequality, 5 < (1/2)a - 7 < a, implies that 'a' must meet both conditions. To understand how to combine these solutions, it's helpful to visualize them on a number line. The inequality a > 24 represents all values of 'a' that lie to the right of 24 on the number line. We can depict this with an open circle at 24 (since 24 is not included) and an arrow extending to the right. Similarly, the inequality a > -14 represents all values of 'a' that lie to the right of -14 on the number line. Again, we use an open circle at -14 and an arrow extending to the right. Now, to find the values of 'a' that satisfy both inequalities, we need to identify the region on the number line where the two solution sets overlap. In other words, we're looking for the intersection of the two sets. Examining the number line, we can see that all values greater than 24 are also greater than -14. However, values between -14 and 24 are not greater than 24. Therefore, the intersection of the two solution sets is the set of all values greater than 24. This means that the solution to the compound inequality is a > 24. In interval notation, this solution is expressed as (24, ∞). This notation indicates that the solution set includes all numbers from 24 (exclusive) to infinity. It's important to recognize that when combining inequalities connected by 'and', we're looking for the overlap, or intersection, of the solution sets. If the inequalities were connected by 'or', we would be looking for the union of the solution sets, which would include all values that satisfy either one inequality or the other (or both). The distinction between 'and' and 'or' is fundamental when working with compound inequalities. In this case, because we have an 'and' situation, the more restrictive inequality determines the overall solution. Since a > 24 is more restrictive than a > -14, it is the solution to the compound inequality. In the final section, we will summarize the steps we took to solve the inequality and provide a concluding statement.

Final Solution

In summary, to solve the compound inequality 5 < (1/2)a - 7 < a, we followed these steps:

1. Isolate the term with 'a' in the middle: We added 7 to all parts of the inequality, resulting in 12 < (1/2)a < a + 7.
2. Eliminate the fraction: We multiplied all parts of the inequality by 2, yielding 24 < a < 2a + 14.
3. Separate the compound inequality: We separated the compound inequality into two individual inequalities: 24 < a and a < 2a + 14.
4. Solve each inequality: We solved each inequality separately, obtaining a > 24 and a > -14.
5. Combine the solutions: We combined the solutions by finding the intersection of the solution sets, which gave us the final solution: a > 24.

Therefore, the solution to the compound inequality 5 < (1/2)a - 7 < a is a > 24, which in interval notation is (24, ∞). This means that any value of 'a' greater than 24 will satisfy the original inequality. Understanding the steps involved in solving compound inequalities is crucial for various mathematical applications. By systematically isolating the variable and combining the solutions of individual inequalities, we can determine the range of values that satisfy the given conditions. It's also essential to remember the underlying principles, such as maintaining the balance of the inequality and correctly interpreting the meaning of 'and' and 'or' when combining solution sets. This problem illustrates a common type of compound inequality that arises in algebra and calculus. The techniques used here can be applied to solve a wide range of similar problems. By practicing these techniques and understanding the underlying concepts, you can develop confidence in your ability to solve inequalities and tackle more complex mathematical challenges. Furthermore, this process highlights the importance of careful algebraic manipulation and attention to detail when working with inequalities. A small error in any step can lead to an incorrect solution. Therefore, it's always a good practice to check your answer by substituting a value from the solution set back into the original inequality to ensure it holds true. This final step of verification can help catch any mistakes and reinforce your understanding of the solution process. Concluding, the detailed step-by-step approach outlined in this article provides a comprehensive guide to solving the compound inequality 5 < (1/2)a - 7 < a, emphasizing both the mechanical steps and the conceptual understanding required for success.