Solving The Inequality 3 < 2a + 9 ≤ 7 A Step-by-Step Guide

In this article, we will delve into the process of solving the compound inequality $3 < 2a + 9 \leqslant 7$. Understanding inequalities is a fundamental concept in mathematics, and this particular example provides a good exercise in manipulating and isolating variables to find the solution set. We will break down each step, providing a clear and concise explanation to ensure you grasp the underlying principles. Solving inequalities is a crucial skill in algebra and calculus, and mastering this topic will pave the way for more advanced mathematical concepts. This guide will not only show you the steps to solve the inequality but also explain the reasoning behind each step, making it easier to apply these techniques to other problems. We will also discuss how to represent the solution graphically, which provides a visual understanding of the solution set. Whether you're a student learning algebra or someone looking to refresh your math skills, this comprehensive guide will provide you with the tools you need to tackle similar problems confidently. The ability to solve inequalities is essential in many areas of mathematics and its applications, including optimization problems, calculus, and real-world scenarios where constraints need to be considered. By the end of this article, you will have a solid understanding of how to approach and solve compound inequalities, a skill that will be invaluable in your mathematical journey.

Understanding Compound Inequalities

Before diving into the specifics of solving $3 < 2a + 9 \leqslant 7$, let's first establish a solid understanding of what compound inequalities are and how they work. Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." In our case, the given inequality $3 < 2a + 9 \leqslant 7$ is a compound inequality that uses an implicit "and." It can be interpreted as two separate inequalities: $3 < 2a + 9$ and $2a + 9 \leqslant 7$. This means that the solution we seek must satisfy both conditions simultaneously. Understanding this separation is crucial because it dictates how we approach the solution process. We need to ensure that any operation we perform on the inequality applies to all parts of it, maintaining the balance and the integrity of the relationship. The "and" condition implies that the solution set will be the intersection of the solutions to each individual inequality. In other words, a value of a must satisfy both $3 < 2a + 9$ and $2a + 9 \leqslant 7$ to be part of the overall solution. This type of inequality is commonly encountered in various mathematical contexts, including optimization problems, interval analysis, and boundary conditions in calculus. Recognizing and understanding the structure of compound inequalities is the first step towards solving them effectively. It's also important to note that "or" compound inequalities, which state that a solution must satisfy at least one of the inequalities, are solved differently. However, in this article, we will focus specifically on the "and" type, as represented by our example.

Step-by-Step Solution: Solving $3 < 2a + 9 \leqslant 7$

To solve the compound inequality $3 < 2a + 9 \leqslant 7$, we need to isolate the variable a in the middle. Isolating the variable is the key to finding the solution set. We will do this by performing the same operations on all three parts of the inequality, maintaining the balance. The first step is to subtract 9 from all parts of the inequality. This will help us to get the term with a by itself. By subtracting 9, we are essentially undoing the addition of 9 in the middle part of the inequality. This gives us:

$3 - 9 < 2a + 9 - 9 \leqslant 7 - 9$

Simplifying this, we get:

$-6 < 2a \leqslant -2$

Now, we have the inequality $-6 < 2a \leqslant -2$. The next step in isolating the variable a is to divide all parts of the inequality by 2. This will remove the coefficient 2 from the term 2a. It's important to remember that when we divide or multiply an inequality by a negative number, we need to flip the direction of the inequality signs. However, in this case, we are dividing by a positive number (2), so the inequality signs remain the same. Dividing all parts by 2, we have:

$\frac{-6}{2} < \frac{2a}{2} \leqslant \frac{-2}{2}$

Simplifying this, we obtain:

$-3 < a \leqslant -1$

This result, $-3 < a \leqslant -1$, represents the solution set for the inequality. It tells us that a is greater than -3 but less than or equal to -1. This is a crucial step in the process, as it directly provides the range of values that a can take to satisfy the original inequality. Understanding this solution is essential for further applications and interpretations. The solution set can also be expressed in interval notation, which is a concise way to represent the range of values. We will discuss interval notation and graphical representation in the following sections to provide a complete understanding of the solution.

Expressing the Solution in Interval Notation

After solving the inequality $-3 < a \leqslant -1$, it's beneficial to express the solution in interval notation. Interval notation is a standard way to represent a set of numbers using intervals, making it easier to visualize and communicate the solution. In this case, the solution $-3 < a \leqslant -1$ means that a can take any value greater than -3 and up to (and including) -1. To write this in interval notation, we use parentheses and brackets. Parentheses ( ) indicate that the endpoint is not included, while brackets [ ] indicate that the endpoint is included. Since a is strictly greater than -3, we use a parenthesis for -3. Since a is less than or equal to -1, we use a bracket for -1. Therefore, the interval notation for the solution is $(-3, -1]$. Understanding interval notation is crucial for various mathematical applications, especially in calculus and analysis. It provides a concise and unambiguous way to represent solution sets, making it easier to work with intervals in different contexts. The left parenthesis in $(-3, -1]$ signifies that -3 is not part of the solution, while the right bracket indicates that -1 is included. This distinction is important in many mathematical operations and applications. Interval notation not only provides a way to represent solutions but also helps in visualizing the solution set on a number line, which we will discuss in the next section. The ability to convert between inequality notation and interval notation is a fundamental skill in mathematics, allowing for a flexible approach to problem-solving and representation.

Graphical Representation of the Solution

To further enhance our understanding of the solution $-3 < a \leqslant -1$, we can represent it graphically on a number line. Graphical representation provides a visual interpretation of the solution set, making it easier to grasp the range of values that a can take. To represent this inequality on a number line, we first draw a horizontal line and mark the relevant points, which in this case are -3 and -1. Since a is strictly greater than -3, we use an open circle at -3 to indicate that -3 is not included in the solution. Since a is less than or equal to -1, we use a closed circle (or a filled-in dot) at -1 to indicate that -1 is included in the solution. Then, we shade the region between -3 and -1 to represent all the values of a that satisfy the inequality. This shaded region, along with the open and closed circles, visually represents the solution set. The graphical representation not only aids in understanding the solution but also serves as a tool for verifying the correctness of the solution. It provides a quick visual check to ensure that the solution includes all the appropriate values and excludes the inappropriate ones. Furthermore, graphical representation is particularly useful when dealing with more complex inequalities or systems of inequalities, as it allows for a clear visualization of the solution region. It also connects the algebraic solution with a geometric interpretation, reinforcing the concepts and making them more intuitive. By visualizing the solution on a number line, we gain a more complete understanding of the range of values that satisfy the original inequality.

Common Mistakes and How to Avoid Them

When solving inequalities, it's crucial to be aware of common mistakes to avoid errors. Avoiding common mistakes is essential for accurate solutions. One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Remember, if you multiply or divide all parts of an inequality by a negative number, you must reverse the direction of the inequality signs. Another common mistake is performing operations on only part of the inequality. In a compound inequality like $3 < 2a + 9 \leqslant 7$, you must apply the same operation to all three parts to maintain the balance. For example, if you subtract 9, you need to subtract it from 3, 2a + 9, and 7. A further mistake arises from incorrect interpretation of the inequality signs. Distinguish between strict inequalities ($<$ and $>$) and inclusive inequalities ($\leqslant$ and $\geqslant$). Strict inequalities use open intervals in interval notation and open circles on the number line, while inclusive inequalities use closed intervals and closed circles. Understanding inequality signs is crucial for correct representation. Another area of potential error is with the order of operations. Always follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions within the inequality. Finally, always check your solution. Substitute a value from your solution set back into the original inequality to ensure it holds true. This step can catch many algebraic errors. By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving inequalities.

Conclusion

In conclusion, solving the inequality $3 < 2a + 9 \leqslant 7$ involves a series of algebraic manipulations to isolate the variable a. We began by subtracting 9 from all parts of the inequality, followed by dividing by 2, resulting in the solution $-3 < a \leqslant -1$. We then expressed this solution in interval notation as $(-3, -1]$ and graphically represented it on a number line using an open circle at -3 and a closed circle at -1, with the region between shaded. This process highlights the importance of understanding compound inequalities and the rules for manipulating them. We also discussed common mistakes to avoid, such as forgetting to flip the inequality sign when multiplying or dividing by a negative number, and the significance of correctly interpreting inequality signs and using appropriate notation. Mastering inequalities is a fundamental skill in mathematics, with applications spanning various fields, including calculus, optimization, and real-world problem-solving. By understanding the concepts and techniques discussed in this article, you are well-equipped to tackle similar problems with confidence. Practice is key to solidifying your understanding, so we encourage you to work through additional examples and explore different types of inequalities. The ability to solve inequalities is not only a valuable mathematical skill but also an essential tool for critical thinking and problem-solving in various contexts. By mastering this topic, you are building a strong foundation for further mathematical studies and real-world applications.