Solving Compound Inequality 5x + 12 ≤ 32 Or 6 + 10x ≥ 96

In mathematics, solving inequalities is a fundamental skill, especially when dealing with compound inequalities. This article provides a comprehensive guide to solving a specific compound inequality involving the 'or' condition. We will break down each step, ensuring a clear understanding of the process. Compound inequalities, which combine two or more inequalities, often require a nuanced approach. These inequalities can be connected by 'and' or 'or,' each requiring a different solution strategy. Here, we will focus on a compound inequality connected by 'or,' specifically the inequality: $5x + 12 ≤ 32$ or $6 + 10x ≥ 96$. Solving this type of inequality involves tackling each part separately and then combining the solutions appropriately. Understanding the 'or' condition is crucial, as it means the solution set includes values that satisfy either inequality, or both. This contrasts with the 'and' condition, where the solution set includes only values that satisfy both inequalities simultaneously. The steps involved in solving compound inequalities are essential for various mathematical applications, including calculus, linear programming, and real-world problem-solving. By mastering these techniques, students and practitioners can better analyze and interpret a wide range of mathematical problems. Let’s embark on the journey of dissecting and solving this compound inequality, gaining insights into the underlying principles and practical steps involved. This comprehensive guide aims to provide a clear, step-by-step approach, making the process accessible to learners of all levels.

Step 1: Solve the First Inequality: 5x + 12 ≤ 32

The first step in addressing this compound inequality is to isolate the variable in each inequality separately. Starting with the first inequality, $5x + 12 ≤ 32$, our goal is to get 'x' alone on one side. This involves using basic algebraic operations to manipulate the inequality while maintaining its balance. The initial step is to subtract 12 from both sides of the inequality. This operation cancels out the +12 on the left side, moving us closer to isolating the term with 'x'. Performing this subtraction, we get $5x + 12 - 12 ≤ 32 - 12$, which simplifies to $5x ≤ 20$. The next step is to divide both sides of the inequality by 5. This will isolate 'x' completely. Dividing by a positive number does not change the direction of the inequality sign, which is an important rule to remember when dealing with inequalities. So, we have $5x / 5 ≤ 20 / 5$, which simplifies to $x ≤ 4$. This result means that all values of 'x' that are less than or equal to 4 satisfy the first inequality. It's crucial to understand that this is just one part of the solution to the entire compound inequality. We still need to solve the second inequality and then combine the solutions appropriately. This step-by-step approach ensures accuracy and clarity in solving complex mathematical problems. By isolating the variable, we can clearly see the range of values that satisfy the individual inequality, which is a critical component in solving the compound inequality as a whole. Understanding this process is fundamental for anyone looking to improve their algebra skills and tackle more advanced mathematical problems.

Step 2: Solve the Second Inequality: 6 + 10x ≥ 96

Now, let's shift our focus to solving the second inequality: $6 + 10x ≥ 96$. Similar to the first inequality, the objective is to isolate the variable 'x'. This requires a series of algebraic manipulations that maintain the inequality's integrity. The first step is to subtract 6 from both sides of the inequality. This will eliminate the constant term on the left side, bringing us closer to isolating the term with 'x'. Performing this operation, we have $6 + 10x - 6 ≥ 96 - 6$, which simplifies to $10x ≥ 90$. The next step involves dividing both sides of the inequality by 10. This operation will completely isolate 'x'. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged. Dividing both sides by 10, we get $10x / 10 ≥ 90 / 10$, which simplifies to $x ≥ 9$. This result indicates that all values of 'x' that are greater than or equal to 9 satisfy the second inequality. It is essential to keep this solution in mind as we proceed to combine it with the solution from the first inequality. Remember, since this is an 'or' compound inequality, the solution set will include all values that satisfy either inequality. This step-by-step process ensures that we accurately determine the range of values that satisfy the second inequality, which is a crucial part of solving the entire compound inequality. Understanding how to isolate the variable and maintain the inequality's balance is fundamental for solving various algebraic problems.

Step 3: Combine the Solutions

Having solved both inequalities, the next crucial step is to combine the solutions. Recall that we have two inequalities: $x ≤ 4$ and $x ≥ 9$. Since the compound inequality is connected by 'or', we are looking for all values of 'x' that satisfy either $x ≤ 4$ or $x ≥ 9$. This means that the solution set includes all numbers less than or equal to 4, as well as all numbers greater than or equal to 9. There is no overlap between these two solution sets, which is perfectly acceptable in an 'or' compound inequality. To visualize this, imagine a number line. The solution $x ≤ 4$ would be represented by a line extending from 4 to the left, including 4. The solution $x ≥ 9$ would be represented by a line extending from 9 to the right, including 9. The entire solution set consists of both of these lines, indicating that any number in either of these ranges is a solution to the compound inequality. In interval notation, we represent this solution as $(−∞, 4] ∪ [9, ∞)$. The symbol '∪' represents the union of the two intervals, meaning we combine all numbers in both intervals into one solution set. Understanding how to combine solutions is essential for solving compound inequalities correctly. The 'or' condition broadens the solution set to include values that satisfy either inequality, as opposed to the 'and' condition, which requires values to satisfy both inequalities simultaneously. This step-by-step approach ensures that we accurately capture the full solution set of the compound inequality.

Step 4: Express the Solution in Interval Notation

To express the solution in interval notation, we need to translate the inequalities $x ≤ 4$ and $x ≥ 9$ into their corresponding interval forms. For $x ≤ 4$, the interval notation is $(−∞, 4]$. This notation indicates that the solution includes all numbers from negative infinity up to and including 4. The parenthesis '(' next to −∞ indicates that negative infinity is not included (as it's a concept, not a number), while the bracket ']' next to 4 indicates that 4 is included in the solution. Similarly, for $x ≥ 9$, the interval notation is $[9, ∞)$. This notation signifies that the solution includes all numbers from 9 up to positive infinity. The bracket '[' next to 9 indicates that 9 is included in the solution, and the parenthesis ')' next to ∞ indicates that positive infinity is not included. Since the compound inequality uses 'or', we combine these two intervals using the union symbol '∪'. Therefore, the final solution in interval notation is $(−∞, 4] ∪ [9, ∞)$. This notation concisely represents the complete solution set, which includes all numbers less than or equal to 4 and all numbers greater than or equal to 9. Interval notation is a standard way to express solutions to inequalities, particularly in higher-level mathematics. It provides a clear and precise representation of the range of values that satisfy the inequality. Understanding how to convert inequalities to interval notation is a crucial skill for anyone studying algebra and beyond. This step ensures that we accurately communicate the solution in a widely recognized mathematical format.

Summary

In summary, we have successfully solved the compound inequality $5x + 12 ≤ 32$ or $6 + 10x ≥ 96$. The process involved several key steps, each crucial for arriving at the correct solution. First, we tackled each inequality separately. For $5x + 12 ≤ 32$, we subtracted 12 from both sides, resulting in $5x ≤ 20$, and then divided by 5 to obtain $x ≤ 4$. Similarly, for $6 + 10x ≥ 96$, we subtracted 6 from both sides, leading to $10x ≥ 90$, and then divided by 10 to get $x ≥ 9$. Next, we combined these individual solutions. Since the compound inequality was connected by 'or', we included all values that satisfy either $x ≤ 4$ or $x ≥ 9$. This means our solution set encompasses all numbers less than or equal to 4 and all numbers greater than or equal to 9. Finally, we expressed this solution in interval notation as $(−∞, 4] ∪ [9, ∞)$. This notation clearly and concisely represents the range of values that satisfy the compound inequality. Understanding how to solve compound inequalities is a fundamental skill in algebra. It requires a careful, step-by-step approach and a clear understanding of the 'or' condition, which allows for solutions that satisfy either inequality. This skill is not only essential for academic success but also for real-world problem-solving where inequalities often arise. By mastering these techniques, students and practitioners can confidently tackle a wide range of mathematical challenges involving inequalities. This comprehensive approach ensures a solid understanding of the process and the underlying mathematical principles.

In conclusion, solving compound inequalities, like the one presented, is a critical skill in mathematics. The process we've outlined – solving each inequality individually and then combining the solutions based on the 'or' condition – provides a clear roadmap for tackling such problems. Understanding these steps not only helps in solving specific mathematical problems but also builds a strong foundation for more advanced mathematical concepts. The ability to manipulate inequalities, isolate variables, and express solutions in various forms, including interval notation, is invaluable in many fields, from engineering and computer science to economics and finance. Moreover, the discipline and logical thinking required to solve inequalities translate to other areas of problem-solving. By mastering these skills, individuals can approach complex challenges with confidence and precision. The 'or' condition in compound inequalities broadens the solution set, requiring a comprehensive understanding of how to combine individual solutions effectively. This knowledge is crucial for accurately interpreting and applying mathematical models in real-world scenarios. As we have demonstrated, a systematic, step-by-step approach is key to success in solving compound inequalities. This involves breaking down the problem into manageable parts, applying the correct algebraic operations, and expressing the solution in a clear and concise manner. The final solution, represented in interval notation, provides a complete picture of the range of values that satisfy the original compound inequality. This comprehensive understanding is essential for anyone looking to excel in mathematics and related fields.