Solving Compound Inequalities A Detailed Guide To 6x + 8 ≤ 20 Or 5 + 4x ≥ 33

Compound inequalities might seem daunting at first, but they are simply two or more inequalities combined. Understanding how to solve them is a fundamental skill in algebra, crucial for various mathematical and real-world applications. In this comprehensive guide, we will dissect a specific compound inequality, $6x + 8 \leq 20$ or $5 + 4x \geq 33$, providing a step-by-step solution and explaining the underlying principles. We'll delve into the logic behind each step, ensuring that you not only understand how to solve it but also why the process works. By the end of this article, you'll be equipped with the knowledge to tackle similar problems with confidence.

Understanding Compound Inequalities

Before diving into the solution, it’s essential to grasp the basics of compound inequalities. A compound inequality is essentially two simple inequalities joined by either "and" or "or." The word "and" indicates that both inequalities must be true simultaneously, while "or" means that at least one of the inequalities must be true. Our example uses "or," which means we’re looking for solutions that satisfy either $6x + 8 \leq 20$ or $5 + 4x \geq 33$, or potentially both. This distinction is critical because it affects how we interpret and combine the solutions. The solutions to an "and" compound inequality are often represented by the intersection of the solutions of the individual inequalities, while the solutions to an "or" compound inequality are represented by the union of the solutions. Think of it this way: "and" narrows down the possibilities, whereas "or" broadens them. Understanding this fundamental difference is the first step in successfully solving compound inequalities.

Step-by-Step Solution of $6x + 8 \leq 20$

Let’s begin by solving the first inequality: $6x + 8 \leq 20$. Our goal is to isolate the variable x on one side of the inequality. To do this, we follow the same principles as solving regular equations, with one crucial difference: when multiplying or dividing by a negative number, we must reverse the direction of the inequality sign. First, we subtract 8 from both sides of the inequality to eliminate the constant term on the left side:

$6x + 8 - 8 \leq 20 - 8$. This simplifies to $6x \leq 12$. Next, we divide both sides by 6 to isolate x:

$\frac{6x}{6} \leq \frac{12}{6}$. This gives us our solution for the first inequality: $x \leq 2$. This means any value of x that is less than or equal to 2 will satisfy this inequality. We can visualize this solution on a number line by shading the region to the left of 2 and including 2 itself (often represented by a closed circle or bracket). This first step is crucial, as it provides one part of the solution set for the overall compound inequality. Missteps in this process, such as forgetting to divide correctly or misinterpreting the inequality sign, can lead to an incorrect final answer. Therefore, careful attention to each step is paramount.

Solving the Second Inequality: $5 + 4x \geq 33$

Now, let’s tackle the second inequality: $5 + 4x \geq 33$. Again, our objective is to isolate x. We begin by subtracting 5 from both sides of the inequality to remove the constant term from the left side:

$5 + 4x - 5 \geq 33 - 5$. This simplifies to $4x \geq 28$. Next, we divide both sides by 4 to isolate x:

$\frac{4x}{4} \geq \frac{28}{4}$. This yields the solution for the second inequality: $x \geq 7$. This signifies that any value of x that is greater than or equal to 7 will satisfy this inequality. Similar to the first inequality, we can represent this solution graphically on a number line by shading the region to the right of 7 and including 7 (again, using a closed circle or bracket). It's vital to remember that we are solving a compound inequality with "or," meaning we need to consider the solutions of both inequalities. Just as with the first inequality, accuracy in each step is crucial. Mistakes in arithmetic or misunderstanding the inequality direction can lead to an incorrect solution set.

Combining the Solutions with "Or"

Now comes the crucial step: combining the solutions of the two inequalities. Remember, the compound inequality is connected by "or," which means we are looking for values of x that satisfy either $x \leq 2$ or $x \geq 7$. The solution set is the union of the two individual solution sets. Visually, this means we take all the values that satisfy $x \leq 2$ (all numbers less than or equal to 2) and all the values that satisfy $x \geq 7$ (all numbers greater than or equal to 7). There is no overlap between these two sets; they are disjointed. This is perfectly acceptable for an "or" compound inequality. The solution set includes all numbers from negative infinity up to and including 2, and all numbers from 7 up to positive infinity. This can be represented in interval notation as $(-\infty, 2] \cup [7, \infty)$. The symbol $\cup$ denotes the union of the two intervals. This final step of combining the solutions is where the understanding of "or" becomes critical. If the connection were "and," we would only consider the values that satisfy both inequalities, which in this case would be an empty set since there is no overlap. The interval notation provides a concise way to represent the solution set, and it is an important skill to master in algebra.

Representing the Solution on a Number Line

Visualizing the solution on a number line is an excellent way to solidify understanding. For the compound inequality $6x + 8 \leq 20$ or $5 + 4x \geq 33$, we have two separate intervals in our solution. The first inequality, $x \leq 2$, is represented on the number line by a closed circle (or a square bracket) at 2 and a line extending to the left, indicating all values less than 2. The second inequality, $x \geq 7$, is represented by a closed circle (or a square bracket) at 7 and a line extending to the right, representing all values greater than 7. Because the inequalities are connected by "or", we include both of these regions in our final solution. The space between 2 and 7 is not included because those values do not satisfy either inequality. The number line provides a clear visual representation of the solution set and helps to avoid errors in interpretation. It reinforces the idea that the solution consists of two distinct intervals and that there are no numbers between 2 and 7 that satisfy the compound inequality. This visual aid is particularly helpful when dealing with more complex compound inequalities or when communicating the solution to others.

Interval Notation for the Solution

As we’ve already touched upon, interval notation is a standard and efficient way to express solution sets, especially for inequalities. For the solution to our compound inequality, $6x + 8 \leq 20$ or $5 + 4x \geq 33$, we have two separate intervals. The first inequality, $x \leq 2$, includes all numbers from negative infinity up to and including 2. In interval notation, this is written as $(-\infty, 2]$. The parenthesis around $-\infty$ indicates that infinity is not a specific number and is never included in the interval. The square bracket around 2 indicates that 2 is included in the interval. The second inequality, $x \geq 7$, includes all numbers from 7 up to positive infinity. In interval notation, this is written as $[7, \infty)$. Again, the square bracket around 7 indicates that 7 is included, and the parenthesis around $\infty$ indicates that infinity is not included. Since the inequalities are connected by "or", we take the union of these two intervals. The union is denoted by the symbol $\cup$. Therefore, the complete solution in interval notation is $(-\infty, 2] \cup [7, \infty)$. Mastering interval notation is essential for expressing solutions concisely and accurately in algebra and beyond. It's a valuable tool for representing sets of numbers and for communicating mathematical ideas effectively.

Common Mistakes to Avoid

Solving compound inequalities, while straightforward in principle, can be prone to errors if care is not taken. One common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This error can completely change the solution set. Another frequent mistake is misinterpreting the meaning of "and" and "or." Remember, "and" requires that both inequalities be satisfied, while "or" requires that at least one be satisfied. Confusing these can lead to an incorrect combination of the individual solutions. A third common mistake is making arithmetic errors during the simplification steps. This highlights the importance of carefully checking each step and ensuring accuracy in calculations. Finally, errors can occur when representing the solution in interval notation or on a number line. It’s crucial to correctly use parentheses and brackets to indicate whether endpoints are included or excluded and to accurately shade the appropriate regions on the number line. By being aware of these common pitfalls and taking the time to work through each step carefully, you can avoid these errors and confidently solve compound inequalities.

Conclusion: Mastering Compound Inequalities

In conclusion, solving compound inequalities like $6x + 8 \leq 20$ or $5 + 4x \geq 33$ is a fundamental skill in algebra. By understanding the basic principles, including the meaning of "and" and "or", and by following a systematic step-by-step approach, you can confidently tackle these problems. We've demonstrated how to solve each inequality individually, how to combine the solutions based on the "or" connector, and how to represent the solution set both on a number line and in interval notation. Moreover, we’ve highlighted common mistakes to avoid, ensuring you approach these problems with accuracy and precision. Mastering compound inequalities not only strengthens your algebraic skills but also provides a foundation for more advanced mathematical concepts. Practice is key to developing fluency, so be sure to work through various examples and apply these techniques to different scenarios. With a solid understanding and consistent practice, you'll be well-equipped to solve compound inequalities and excel in your mathematical endeavors. Remember, each step builds upon the previous one, so clarity and precision at each stage are crucial for arriving at the correct solution. Keep practicing, and you’ll find these types of problems become second nature.