Solving Compound Inequalities A Step-by-Step Guide

In the realm of mathematics, compound inequalities present a unique challenge, requiring us to consider two or more inequalities simultaneously. These inequalities are connected by the words "and" or "or," each dictating a different approach to finding the solution set. In this comprehensive guide, we will delve into the intricacies of solving compound inequalities, focusing on the specific example: $4y - 1 \\\leq -25$ or $-3y \\\leq -12$. We will explore the step-by-step process, the underlying principles, and the nuances of expressing the solution in interval notation. Understanding compound inequalities is crucial for various mathematical applications, from calculus to optimization problems. This guide aims to equip you with the knowledge and skills necessary to confidently tackle these types of problems.

Understanding Compound Inequalities

Before diving into the solution, it's essential to grasp the fundamental concept of compound inequalities. A compound inequality is essentially two simple inequalities combined into one statement using either the word "and" or the word "or." The word "and" signifies an intersection, meaning the solution must satisfy both inequalities simultaneously. In contrast, the word "or" indicates a union, meaning the solution must satisfy at least one of the inequalities. This distinction is crucial as it dictates how we interpret and solve the problem. For instance, consider the compound inequality "x > 3 and x < 7." The solution includes all values of x that are greater than 3 and less than 7, effectively the numbers between 3 and 7, not including 3 and 7. On the other hand, the compound inequality "x < 2 or x > 5" includes all values of x that are less than 2 or greater than 5. This means we have two separate intervals that form the solution. Recognizing whether a compound inequality uses "and" or "or" is the first step in determining the appropriate solution strategy.

Solving the First Inequality: $4y - 1 \\\\\leq -25$

Let's begin by tackling the first inequality: $4y - 1 \\\leq -25$. Our goal is to isolate the variable y on one side of the inequality. We can achieve this by performing a series of algebraic operations while maintaining the inequality's balance. The first step involves adding 1 to both sides of the inequality. This eliminates the constant term on the left side, bringing us closer to isolating y. Adding 1 to both sides gives us: $4y - 1 + 1 \\\leq -25 + 1$, which simplifies to $4y \\\leq -24$. Now, we have a simpler inequality. To completely isolate y, we need to eliminate the coefficient 4. This can be done by dividing both sides of the inequality by 4. It is important to remember that when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. However, in this case, we are dividing by a positive number (4), so the inequality sign remains unchanged. Dividing both sides by 4, we get: $(4y)/4 \\\leq (-24)/4$, which simplifies to $y \\\leq -6$. This is the solution to the first inequality. It tells us that y can be any value less than or equal to -6. This is a critical component of the overall solution to the compound inequality. Now that we've solved the first inequality, we move on to the second inequality, keeping in mind that the "or" connector means we're looking for values of y that satisfy either this inequality or the previous one.

Solving the Second Inequality: $-3y \\\leq -12$

Now, let's address the second inequality: $-3y \\\leq -12$. Similar to the first inequality, our objective is to isolate the variable y. However, this time we encounter a crucial difference: the coefficient of y is negative (-3). This means that when we divide both sides of the inequality by -3, we must reverse the direction of the inequality sign. This is a fundamental rule in inequality manipulations, and failing to apply it correctly will lead to an incorrect solution. To isolate y, we divide both sides of the inequality by -3: $(-3y)/(-3) \\\geq (-12)/(-3)$. Notice that the inequality sign has flipped from "less than or equal to" to "greater than or equal to." This is the key step. Simplifying the division, we get: $y \\\geq 4$. This is the solution to the second inequality. It tells us that y can be any value greater than or equal to 4. This solution, combined with the solution from the first inequality, will form the complete solution set for the compound inequality. The "or" connector means we're looking for values of y that satisfy either $y \\\leq -6$ or $y \\\geq 4$.

Combining the Solutions with "Or"

Having solved both inequalities, we now need to combine the solutions. The compound inequality is connected by the word "or," which means we are looking for the union of the two solution sets. In other words, we want all values of y that satisfy either $y \\\leq -6$ or $y \\\geq 4$. The solution $y \\\leq -6$ represents all numbers less than or equal to -6, while the solution $y \\\geq 4$ represents all numbers greater than or equal to 4. There is no overlap between these two solution sets. The numbers between -6 and 4 are not included in the solution. To visualize this, imagine a number line. One part of the solution is a ray extending to the left from -6 (including -6), and the other part is a ray extending to the right from 4 (including 4). There is a gap in the middle. This gap is what distinguishes an "or" compound inequality from an "and" compound inequality, where we would be looking for the intersection of the solution sets. Because there is no overlap, the solution to this compound inequality consists of two separate intervals. The next step is to express this solution in interval notation.

Expressing the Solution in Interval Notation

To express the solution in interval notation, we use brackets and parentheses to indicate whether the endpoints are included or excluded, and we use the union symbol "∪" to connect the two intervals. The solution $y \\\leq -6$ includes all numbers from negative infinity up to and including -6. In interval notation, this is written as $(-∞, -6]$. The parenthesis on the left indicates that negative infinity is not a specific number and is always excluded. The square bracket on the right indicates that -6 is included in the solution. The solution $y \\\geq 4$ includes all numbers from 4 up to positive infinity. In interval notation, this is written as $[4, ∞)$. The square bracket on the left indicates that 4 is included in the solution, and the parenthesis on the right indicates that positive infinity is always excluded. Since we are looking for the union of these two intervals (because the original compound inequality used "or"), we combine these two interval notations using the union symbol: $(-∞, -6] ∪ [4, ∞)$. This is the final solution to the compound inequality expressed in interval notation. It concisely represents all the values of y that satisfy the original inequality.

In summary, solving compound inequalities involves a series of steps: first, understanding the role of the connecting words "and" or "or," then solving each inequality separately, and finally, combining the solutions appropriately. For the compound inequality $4y - 1 \\\leq -25$ or $-3y \\\leq -12$, we found that the solution is $y \\\leq -6$ or $y \\\geq 4$. Expressing this in interval notation, we get $(-∞, -6] ∪ [4, ∞)$. This final answer represents the set of all values that satisfy the original compound inequality. Mastering the techniques for solving compound inequalities is a fundamental skill in algebra, with applications in various mathematical fields and real-world problems. By carefully following the steps outlined in this guide, you can confidently tackle compound inequalities and express their solutions accurately.