Solving Inequalities 5x - 4 ≥ 12 OR 12x + 5 ≤ -4 A Step-by-Step Guide

Mastering inequalities is a fundamental skill in algebra, with applications spanning various fields of mathematics and beyond. In this comprehensive guide, we will delve into the step-by-step solution of the compound inequality $5x - 4 \geq 12$ OR $12x + 5 \leq -4$, providing a clear and concise explanation of the underlying principles and techniques. This guide aims to equip you with the knowledge and confidence to tackle similar problems effectively. Understanding inequalities is crucial for solving a wide range of mathematical problems, from determining the range of possible solutions to optimizing real-world scenarios. The ability to manipulate and solve inequalities is a cornerstone of algebraic proficiency, opening doors to more advanced mathematical concepts. Whether you are a student learning algebra for the first time or a seasoned mathematician looking for a refresher, this guide will offer valuable insights and practical strategies. We will explore the properties of inequalities, the rules for manipulating them, and the methods for representing their solutions graphically and in interval notation. By the end of this guide, you will have a solid understanding of how to solve compound inequalities and apply these skills to a variety of problems.

Understanding the Basics of Inequalities

Before we dive into the solution, let's establish a solid foundation by reviewing the basics of inequalities. Inequalities are mathematical statements that compare two expressions using symbols such as greater than ($>$), less than ($<$), greater than or equal to ($\geq$), and less than or equal to ($\leq$). Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of solutions. This range represents all the values that satisfy the inequality. For instance, the inequality $x > 3$ means that any value of $x$ greater than 3 is a solution. This includes numbers like 3.01, 4, 5, and so on. The concept of a range of solutions is crucial to understanding the difference between equations and inequalities. Inequalities are used to describe situations where a quantity is not exactly equal to another but falls within a certain range. This is particularly useful in real-world applications, where exact values may not be known or may vary within a certain margin. For example, in engineering, inequalities might be used to define the acceptable range of values for a particular parameter. In economics, they might be used to model the range of possible outcomes in a market. The ability to work with inequalities is therefore an essential skill for anyone dealing with quantitative data.

Properties of Inequalities

Just like equations, inequalities have certain properties that allow us to manipulate them while preserving their truth. These properties are essential for solving inequalities algebraically. One of the most important properties is the addition property of inequality, which states that adding or subtracting the same number from both sides of an inequality does not change its direction. For example, if we have the inequality $x - 2 < 5$, we can add 2 to both sides to get $x < 7$. Similarly, the multiplication property of inequality states that multiplying or dividing both sides of an inequality by a positive number does not change its direction. However, if we multiply or divide both sides by a negative number, we must reverse the direction of the inequality sign. For example, if we have the inequality $-2x > 6$, we divide both sides by -2, and we need to reverse the inequality sign, resulting in $x < -3$. These properties are fundamental to solving inequalities, and understanding them is crucial for avoiding common mistakes. It's important to remember the rule about reversing the inequality sign when multiplying or dividing by a negative number, as this is a frequent source of errors. By applying these properties correctly, we can isolate the variable and find the solution set of the inequality.

Solving the First Inequality: $5x - 4 \geq 12$

Let's start by solving the first inequality, $5x - 4 \geq 12$. The goal is to isolate $x$ on one side of the inequality. We can achieve this by performing a series of algebraic operations. First, we add 4 to both sides of the inequality, applying the addition property of inequality. This gives us: $5x - 4 + 4 \geq 12 + 4$ which simplifies to $5x \geq 16$. Next, we divide both sides of the inequality by 5, applying the multiplication property of inequality. Since 5 is a positive number, we don't need to reverse the inequality sign. This gives us: $\frac{5x}{5} \geq \frac{16}{5}$ which simplifies to $x \geq \frac{16}{5}$. This means that any value of $x$ that is greater than or equal to $\frac{16}{5}$ will satisfy the first inequality. To visualize this solution, we can imagine a number line. The solution to $x \geq \frac{16}{5}$ would be represented by a closed circle at $\frac{16}{5}$ (indicating that $\frac{16}{5}$ is included in the solution) and a line extending to the right, representing all values greater than $\frac{16}{5}$. Understanding how to isolate the variable in an inequality is a key skill in algebra. By carefully applying the properties of inequalities, we can systematically solve for $x$ and determine the solution set.

Solving the Second Inequality: $12x + 5 \leq -4$

Now, let's tackle the second inequality, $12x + 5 \leq -4$. Similar to the first inequality, our aim is to isolate $x$. We begin by subtracting 5 from both sides of the inequality, using the addition property of inequality. This yields: $12x + 5 - 5 \leq -4 - 5$ which simplifies to $12x \leq -9$. Next, we divide both sides of the inequality by 12, applying the multiplication property of inequality. Since 12 is a positive number, we don't need to reverse the inequality sign. This gives us: $\frac{12x}{12} \leq \frac{-9}{12}$ which simplifies to $x \leq -\frac{3}{4}$. This means that any value of $x$ that is less than or equal to $-\frac{3}{4}$ will satisfy the second inequality. Visualizing this solution on a number line, we would have a closed circle at $-\frac{3}{4}$ (indicating that $-\frac{3}{4}$ is included in the solution) and a line extending to the left, representing all values less than $-\frac{3}{4}$. The process of isolating the variable in this inequality demonstrates the importance of applying the properties of inequalities carefully. By following the correct steps, we can arrive at the solution set and understand the range of values that satisfy the inequality.

Combining the Solutions: "OR" Compound Inequality

The original problem presents a compound inequality connected by the word "OR". This means that the solution set includes all values of $x$ that satisfy either the first inequality or the second inequality. In other words, we are looking for the union of the solution sets of the two inequalities. We found that the solution to the first inequality, $5x - 4 \geq 12$, is $x \geq \frac{16}{5}$. The solution to the second inequality, $12x + 5 \leq -4$, is $x \leq -\frac{3}{4}$. To combine these solutions, we simply take all values that satisfy either inequality. This means that the solution to the compound inequality $5x - 4 \geq 12$ OR $12x + 5 \leq -4$ is $x \geq \frac{16}{5}$ or $x \leq -\frac{3}{4}$. On a number line, this would be represented by two separate intervals: one extending from $-\infty$ to $-\frac{3}{4}$, including $-\frac{3}{4}$, and another extending from $\frac{16}{5}$ to $\infty$, including $\frac{16}{5}$. The "OR" in the compound inequality signifies that any value within either of these intervals is a solution. Understanding how to combine solutions in compound inequalities is crucial for solving more complex problems. The "OR" condition creates a union of solution sets, while an "AND" condition would create an intersection of solution sets.

The Final Answer and Its Significance

Therefore, the final answer to the compound inequality $5x - 4 \geq 12$ OR $12x + 5 \leq -4$ is $x \geq \frac{16}{5}$ or $x \leq -\frac{3}{4}$. This corresponds to answer choice A. This solution set includes all values of $x$ that are either greater than or equal to $\frac{16}{5}$ or less than or equal to $-\frac{3}{4}$. It's important to understand the significance of this solution. It tells us that there are infinitely many values of $x$ that satisfy the given compound inequality. These values fall into two distinct intervals, separated by a gap. This type of solution is common in compound inequalities involving "OR", where the solution sets of the individual inequalities may not overlap. Understanding the solution set in this way is crucial for applying these concepts to real-world problems. For example, if we were modeling a physical system, the solution set might represent the range of values for a particular parameter that would allow the system to function correctly. The ability to solve compound inequalities is therefore an essential tool for anyone working with mathematical models and quantitative data. The solution $x \geq \frac{16}{5}$ or $x \leq -\frac{3}{4}$ highlights the power of inequalities in describing a range of possible values, rather than just a single value.

Conclusion

In conclusion, solving the compound inequality $5x - 4 \geq 12$ OR $12x + 5 \leq -4$ involves a series of steps, each grounded in the fundamental properties of inequalities. By carefully isolating the variable in each inequality and then combining the solutions using the "OR" condition, we arrive at the final answer: $x \geq \frac{16}{5}$ or $x \leq -\frac{3}{4}$. This solution represents a range of values that satisfy the given compound inequality. Mastering the techniques for solving inequalities is a crucial skill in algebra, with applications in various fields. By understanding the properties of inequalities and practicing problem-solving, you can develop the confidence and expertise to tackle even more complex mathematical challenges. Inequalities are not just abstract mathematical concepts; they are powerful tools for modeling and solving real-world problems. From engineering and economics to computer science and physics, inequalities play a vital role in describing and analyzing quantitative relationships. Therefore, investing time in understanding and mastering inequalities is an investment in your mathematical and problem-solving skills that will pay dividends in many areas of your life and career. The ability to think critically about inequalities and their solutions will empower you to make informed decisions and solve complex problems in a variety of contexts. Keep practicing, and you'll become a master of inequalities!