Solving Inequalities A Step-by-Step Guide To 5a + 9 ≤ 2a + 15

Introduction

In the realm of mathematics, inequalities play a crucial role in expressing relationships where quantities are not necessarily equal. Inequalities are mathematical statements that compare two expressions using symbols such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Solving inequalities involves finding the range of values that satisfy the given condition, much like solving equations but with a few key differences. This article delves into the step-by-step process of solving the linear inequality 5a + 9 ≤ 2a + 15, providing a comprehensive understanding of the underlying principles and techniques. The process involves isolating the variable 'a' on one side of the inequality to determine the set of values that make the inequality true. Understanding how to manipulate inequalities is fundamental in various mathematical contexts, including algebra, calculus, and real analysis. Furthermore, the skills acquired in solving inequalities are applicable to real-world problem-solving scenarios where constraints and limitations need to be considered. This article aims to equip readers with the knowledge and confidence to tackle similar inequality problems with ease. By mastering the techniques presented here, readers will be able to approach more complex mathematical challenges and apply these concepts to practical situations. The step-by-step breakdown ensures that each stage of the solution is clear and understandable, making the process accessible to learners of all levels. From the initial setup to the final solution, every aspect of solving the inequality is explained in detail, reinforcing the importance of precision and accuracy in mathematical manipulations. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this guide offers a valuable resource for understanding and solving linear inequalities.

Step 1: Understanding the Inequality

Before diving into the solution, it's crucial to understand what the inequality 5a + 9 ≤ 2a + 15 represents. This mathematical statement asserts that the expression 5a + 9 is less than or equal to the expression 2a + 15. The goal is to find all values of 'a' that make this statement true. Inequalities are similar to equations, but instead of seeking a single solution, we are looking for a range of solutions. The symbol '≤' signifies that the left-hand side can be equal to or less than the right-hand side. This distinction is important because it affects how we interpret the final answer. Understanding the nature of the inequality helps in visualizing the solution set, which could be a range of numbers rather than a single point. In practical terms, inequalities are used to model situations where there are constraints or limitations, such as budget constraints, resource limitations, or physical boundaries. For example, an inequality might represent the maximum number of items that can be produced with a given amount of resources, or the range of temperatures within which a device can operate safely. The ability to interpret and solve inequalities is therefore a valuable skill in many fields, including economics, engineering, and computer science. In this specific inequality, 5a + 9 ≤ 2a + 15, we are comparing two linear expressions. Each expression represents a straight line, and the inequality asks for the values of 'a' where the line represented by 5a + 9 lies below or coincides with the line represented by 2a + 15. This graphical interpretation provides a visual aid to understanding the solution. By mastering the art of interpreting inequalities, one can apply them effectively in various mathematical and real-world contexts.

Step 2: Isolating the Variable 'a'

To solve for 'a', the next step is to isolate the variable on one side of the inequality. This is achieved by performing algebraic operations on both sides while maintaining the inequality's balance. The first operation involves subtracting 2a from both sides of the inequality: 5a + 9 - 2a ≤ 2a + 15 - 2a. This simplifies to 3a + 9 ≤ 15. Subtracting 2a from both sides helps to consolidate the terms involving 'a' on one side, making it easier to isolate the variable. This step is crucial because it reduces the complexity of the inequality and brings us closer to the solution. It is important to perform the same operation on both sides to preserve the validity of the inequality. Next, we need to isolate the term with 'a' further. This is done by subtracting 9 from both sides: 3a + 9 - 9 ≤ 15 - 9. This simplifies to 3a ≤ 6. Subtracting 9 from both sides removes the constant term on the left side, leaving only the term involving 'a'. This step is a standard algebraic technique used to isolate variables in equations and inequalities. The key principle here is to perform inverse operations to undo the operations that are applied to the variable. In this case, we subtracted 9 to undo the addition of 9. The goal of isolating the variable is to get 'a' by itself, which will allow us to determine the range of values that satisfy the inequality. Each step in this process is carefully chosen to move us closer to this goal while ensuring that the inequality remains balanced. By mastering the techniques of isolating variables, one can solve a wide variety of algebraic problems.

Step 3: Solving for 'a'

After isolating the term with 'a', the final step is to solve for 'a' by dividing both sides of the inequality by the coefficient of 'a'. In this case, the inequality is 3a ≤ 6. To isolate 'a', we divide both sides by 3: (3a) / 3 ≤ 6 / 3. This simplifies to a ≤ 2. Dividing both sides by 3 gives us the solution for 'a', which is a range of values rather than a single value. The important rule to remember when solving inequalities is that if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. However, since we are dividing by a positive number (3), the inequality sign remains the same. The solution a ≤ 2 means that any value of 'a' that is less than or equal to 2 will satisfy the original inequality 5a + 9 ≤ 2a + 15. This includes numbers such as 2, 1, 0, -1, -2, and so on. The solution set is an infinite range of values, which is a common characteristic of inequalities. To visualize the solution, one can think of a number line. The solution a ≤ 2 represents all the points on the number line that are to the left of and including the point 2. This visual representation helps in understanding the range of possible values for 'a'. The process of solving for 'a' involves careful application of algebraic rules to maintain the balance of the inequality while isolating the variable. Each step is crucial, and accuracy is essential to arrive at the correct solution. Understanding the concept of inequalities and how to solve them is a fundamental skill in algebra and beyond. It allows one to model and solve problems involving constraints and limitations, which are common in various fields such as economics, engineering, and computer science.

Step 4: Verifying the Solution

To ensure the solution a ≤ 2 is correct, it's essential to verify it by substituting values from the solution set back into the original inequality. This process helps confirm that the solution set indeed satisfies the given condition. First, let's test the boundary value, a = 2. Substituting a = 2 into the original inequality 5a + 9 ≤ 2a + 15, we get: 5(2) + 9 ≤ 2(2) + 15, which simplifies to 10 + 9 ≤ 4 + 15, and further to 19 ≤ 19. This statement is true, as 19 is equal to 19, confirming that a = 2 is part of the solution set. Next, let's test a value less than 2, say a = 0. Substituting a = 0 into the original inequality, we get: 5(0) + 9 ≤ 2(0) + 15, which simplifies to 9 ≤ 15. This statement is also true, as 9 is less than 15, indicating that values less than 2 are also part of the solution set. Now, let's test a value greater than 2, say a = 3. Substituting a = 3 into the original inequality, we get: 5(3) + 9 ≤ 2(3) + 15, which simplifies to 15 + 9 ≤ 6 + 15, and further to 24 ≤ 21. This statement is false, as 24 is not less than or equal to 21, confirming that values greater than 2 are not part of the solution set. These tests provide strong evidence that the solution a ≤ 2 is correct. Verification is a crucial step in solving any mathematical problem, as it helps to catch any errors that may have occurred during the solution process. By substituting values back into the original inequality, we can gain confidence that our solution is accurate. This step is particularly important in inequalities, where the solution is a range of values rather than a single value. The verification process reinforces the understanding of the inequality and the solution set, ensuring a thorough grasp of the concept.

Step 5: Expressing the Solution

After solving and verifying the inequality, the final step is to express the solution in a clear and concise manner. The solution a ≤ 2 can be expressed in several ways, depending on the context and the desired level of formality. One way to express the solution is in set notation. The solution set can be written as {a | a ≤ 2}, which reads as "the set of all 'a' such that 'a' is less than or equal to 2." This notation is precise and commonly used in mathematical literature. Another way to express the solution is graphically on a number line. Draw a number line and mark the point 2. Since the inequality includes the 'equal to' condition (≤), we use a closed circle (or a filled-in dot) at the point 2 to indicate that 2 is included in the solution set. Then, shade the region to the left of 2, representing all the values less than 2. This visual representation provides an intuitive understanding of the solution set. The solution can also be expressed in interval notation. The interval notation for a ≤ 2 is (-∞, 2]. The parenthesis '(' indicates that the endpoint is not included in the interval, while the bracket ']' indicates that the endpoint is included. The symbol '-∞' represents negative infinity, indicating that the interval extends indefinitely to the left. Interval notation is a compact and convenient way to represent solution sets, especially for more complex inequalities. Expressing the solution in multiple ways helps to reinforce understanding and allows for flexibility in different contexts. Whether using set notation, graphical representation, or interval notation, the goal is to communicate the solution clearly and accurately. The choice of notation may depend on the audience and the specific requirements of the problem. By mastering the different ways to express solutions, one can effectively communicate mathematical results and apply them in various fields.

Conclusion

In conclusion, solving the inequality 5a + 9 ≤ 2a + 15 involves a series of algebraic steps, each crucial for arriving at the correct solution. The process begins with understanding the inequality, recognizing that it represents a range of values rather than a single solution. Next, the variable 'a' is isolated by performing operations on both sides of the inequality, maintaining its balance. This involves subtracting terms to consolidate 'a' on one side and then dividing by the coefficient of 'a' to solve for 'a'. The solution obtained, a ≤ 2, indicates that any value of 'a' less than or equal to 2 will satisfy the original inequality. To ensure accuracy, the solution is verified by substituting values from the solution set back into the original inequality. This step confirms that the solution is correct and helps to catch any potential errors. Finally, the solution is expressed in a clear and concise manner, using set notation, graphical representation on a number line, or interval notation. Each method offers a different perspective on the solution set, enhancing understanding and communication. The ability to solve inequalities is a fundamental skill in mathematics, with applications in various fields such as economics, engineering, and computer science. Mastering this skill requires a thorough understanding of algebraic principles and careful attention to detail. By following the step-by-step approach outlined in this article, readers can confidently tackle similar inequality problems and apply these concepts to real-world scenarios. Inequalities are not just abstract mathematical concepts; they are powerful tools for modeling and solving problems involving constraints and limitations. Whether you are a student learning algebra or a professional applying mathematical techniques, the ability to solve inequalities is an invaluable asset.