Solving -10 < 6x + 2 ≤ 32 A Step-by-Step Guide
In the realm of mathematics, solving inequalities is a fundamental skill. This article provides a detailed walkthrough of solving the compound inequality -10 < 6x + 2 ≤ 32. We will break down each step, ensuring a clear understanding of the process. This problem falls under the category of algebra, specifically dealing with linear inequalities. Mastering this type of problem is crucial for more advanced mathematical concepts. The ability to solve inequalities is essential not only for academic success but also for various real-world applications. From determining budget constraints to optimizing resource allocation, inequalities play a vital role in problem-solving across different domains. In this guide, we will not only solve the given inequality but also discuss the underlying principles and concepts, empowering you to tackle similar problems with confidence.
Understanding Inequalities
Before diving into the solution, it's crucial to understand what inequalities represent. Inequalities, unlike equations, do not have a single solution but rather a range of solutions. The symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) are used to express these relationships. A compound inequality, such as the one we're dealing with, combines two or more inequalities into a single statement. In our case, -10 < 6x + 2 ≤ 32 signifies that the expression 6x + 2 is greater than -10 and simultaneously less than or equal to 32. This means the solution set will include all values of x that satisfy both conditions. Visualizing inequalities on a number line can be helpful. For instance, x > 3 represents all points to the right of 3 (excluding 3 itself), while x ≤ 5 includes all points to the left of 5, including 5. Understanding these fundamental concepts is essential for accurately interpreting and solving inequalities. Without a solid grasp of these basics, it becomes challenging to manipulate inequalities and arrive at the correct solution. The ability to translate real-world scenarios into mathematical inequalities is also a valuable skill that stems from this understanding.
Step-by-Step Solution: -10 < 6x + 2 ≤ 32
To solve the compound inequality -10 < 6x + 2 ≤ 32, we need to isolate the variable x in the middle. This involves performing the same operations on all three parts of the inequality to maintain the balance. Our first step is to eliminate the constant term (+2) from the middle. We achieve this by subtracting 2 from all parts of the inequality:
-10 - 2 < 6x + 2 - 2 ≤ 32 - 2. This simplifies to -12 < 6x ≤ 30. Next, we need to isolate x by dividing all parts of the inequality by the coefficient of x, which is 6: (-12) / 6 < (6x) / 6 ≤ 30 / 6. This simplifies to -2 < x ≤ 5. Therefore, the solution to the compound inequality is -2 < x ≤ 5. This means that x can take any value greater than -2 but less than or equal to 5. It's important to note that dividing by a negative number would require flipping the inequality signs, but in this case, we are dividing by a positive number, so the signs remain the same. This step-by-step approach ensures clarity and reduces the chances of making errors. By breaking down the problem into smaller, manageable steps, we can effectively solve even complex inequalities. Furthermore, understanding the rationale behind each step is crucial for applying these techniques to other problems.
Interpreting the Solution
The solution -2 < x ≤ 5 tells us that x can be any number greater than -2 and less than or equal to 5. This range includes all numbers between -2 and 5, but it excludes -2 and includes 5. We can represent this solution graphically on a number line. On a number line, we would use an open circle at -2 to indicate that it is not included in the solution and a closed circle at 5 to indicate that it is included. The region between these two points is shaded to represent all the possible values of x. This visual representation helps to solidify the understanding of the solution set. The solution can also be expressed in interval notation. In this notation, we use parentheses for values that are not included and brackets for values that are included. Therefore, the solution -2 < x ≤ 5 can be written in interval notation as (-2, 5]. This notation is a concise and commonly used way to represent the solution set of inequalities. Understanding how to interpret and represent solutions in different forms is essential for effective communication in mathematics. Whether it's graphical representation, interval notation, or simply stating the solution in words, clarity and accuracy are paramount.
Common Mistakes and How to Avoid Them
When solving inequalities, several common mistakes can lead to incorrect solutions. One frequent error is forgetting to flip the inequality sign when multiplying or dividing by a negative number. For example, if we have -2x < 4, dividing both sides by -2 requires flipping the sign to get x > -2. Another common mistake is incorrectly distributing a negative sign when dealing with expressions in parentheses. For instance, - (x + 3) should be distributed as -x - 3, not -x + 3. Failing to perform the same operation on all parts of a compound inequality is another pitfall. Remember, to maintain the balance, any operation performed on one part must be applied to all parts. Additionally, misinterpreting the inequality symbols can lead to errors. Pay close attention to whether the symbol is < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). To avoid these mistakes, it's crucial to practice consistently and carefully review each step of the solution. Double-checking your work and seeking clarification when needed are also valuable strategies. By being aware of these common pitfalls and implementing preventative measures, you can significantly improve your accuracy in solving inequalities. Developing a systematic approach to problem-solving, including checking your work and understanding the underlying principles, will further minimize the chances of making errors.
Real-World Applications of Inequalities
Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. In finance, inequalities can be used to represent budget constraints. For example, if you have a budget of $100 and want to buy two items, one costing $30 and the other costing x, the inequality 30 + x ≤ 100 represents the possible values of x you can afford. In business, inequalities can help optimize resource allocation. For instance, a company might want to maximize its profit while staying within certain production constraints. This can be modeled using a system of inequalities and solved using linear programming techniques. In engineering, inequalities are used to ensure safety and performance. For example, the load capacity of a bridge must be greater than or equal to the expected load, which can be expressed as an inequality. In everyday life, inequalities can be used to make informed decisions. For instance, if you need to drive a certain distance in a limited time, you can use inequalities to calculate the minimum speed you need to maintain. These examples highlight the practical significance of inequalities in various fields. Understanding how to apply inequalities to real-world problems can help you make better decisions and solve complex challenges. The ability to translate real-world scenarios into mathematical models involving inequalities is a valuable skill that is applicable across many disciplines.
Practice Problems
To solidify your understanding of solving inequalities, let's consider some practice problems. Try solving the following inequalities on your own:
- 2x - 5 > 3
- -3x + 1 ≤ 10
- -15 < 5x + 5 ≤ 20
- 4(x - 2) ≥ 8
- -2 < (4x - 6) / 2 < 5
For each problem, remember to isolate the variable by performing the same operations on both sides of the inequality. Pay attention to the inequality signs and remember to flip the sign if you multiply or divide by a negative number. Once you've found the solution, try representing it on a number line and in interval notation. Working through these practice problems will help you develop confidence and proficiency in solving inequalities. It's also beneficial to check your answers and review the steps if you encounter any difficulties. Consistent practice is key to mastering mathematical concepts, and inequalities are no exception. The more you practice, the more comfortable you will become with the different types of problems and the various techniques used to solve them. Consider seeking out additional practice problems from textbooks, online resources, or your instructor to further enhance your skills.
By understanding the concepts, following the steps, avoiding common mistakes, and practicing regularly, you can master the skill of solving inequalities and apply it effectively in various contexts. Remember, mathematics is a building block, and each concept you learn builds upon the previous ones. A strong foundation in inequalities will undoubtedly benefit you in your future mathematical endeavors.
