Solving Inequalities A Step-by-Step Guide To 8(x-5)-2 ≥ -20

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In this article, we will delve into the process of solving linear inequalities, using the example of 8(x-5)-2 ≥ -20 as our guide. Understanding how to solve inequalities is crucial in various fields, from mathematics and science to economics and engineering. This article aims to provide a comprehensive, step-by-step approach to solving linear inequalities, ensuring clarity and understanding for learners of all levels. We'll break down each step, providing explanations and justifications along the way. We will explore the properties of inequalities and how they differ from equations, and finally learn how to represent the solution set graphically and in interval notation. By the end of this article, you will not only be able to solve this specific inequality but also have a solid foundation for tackling other similar problems.

Understanding Inequalities

Before we dive into the solution, let's first understand what inequalities are. Unlike equations, which show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The basic inequality symbols are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Solving an inequality means finding the range of values that satisfy the given relationship. This range of values is known as the solution set. Inequalities are fundamental in mathematics and are used to model real-world situations where exact values are not possible or necessary. For example, an inequality might represent a budget constraint, a speed limit, or the minimum number of items needed to achieve a certain goal. Understanding inequalities is essential for problem-solving in various fields, including economics, engineering, and computer science. The rules for solving inequalities are similar to those for solving equations, with one crucial difference: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This difference is critical to keep in mind to obtain the correct solution set.

Step 1: Distribute

The first step in solving the inequality 8(x-5)-2 ≥ -20 is to simplify the expression by distributing the 8 across the terms inside the parentheses. This involves multiplying 8 by both x and -5. This gives us: 8 * x - 8 * 5 - 2 ≥ -20, which simplifies to 8x - 40 - 2 ≥ -20. Distribution is a fundamental algebraic operation that allows us to remove parentheses and combine like terms. It's important to perform this step accurately, as any errors here will propagate through the rest of the solution. The distributive property states that a(b + c) = ab + ac. In this case, a = 8, b = x, and c = -5. After distributing, we have a simpler expression that is easier to manipulate and solve. This step is crucial for isolating the variable and finding the solution set. Distribution is not only essential for solving inequalities but also for simplifying algebraic expressions in general. It allows us to combine like terms and make expressions easier to work with. By mastering distribution, you'll be well-equipped to tackle more complex algebraic problems.

Step 2: Combine Like Terms

Now that we've distributed, we can simplify the inequality further by combining like terms. In the expression 8x - 40 - 2 ≥ -20, we have two constant terms, -40 and -2, which can be combined. Adding these terms together gives us: 8x - 42 ≥ -20. Combining like terms is a crucial step in simplifying algebraic expressions and equations. It allows us to reduce the number of terms and make the expression more manageable. In this case, combining the constant terms makes it easier to isolate the variable in the next steps. Like terms are terms that have the same variable raised to the same power. Constant terms, such as -40 and -2, are also considered like terms because they do not contain any variables. Combining like terms involves adding or subtracting their coefficients. In this case, we added -40 and -2 to get -42. By combining like terms, we've simplified the inequality and brought it closer to the standard form for solving linear inequalities. This step is essential for efficiently isolating the variable and finding the solution set. Neglecting to combine like terms can lead to unnecessary complexity and increase the chances of making errors in subsequent steps.

Step 3: Isolate the Variable Term

Our next goal is to isolate the term containing the variable, which in this case is 8x. To do this, we need to get rid of the constant term on the same side of the inequality, which is -42. We can achieve this by adding 42 to both sides of the inequality: 8x - 42 + 42 ≥ -20 + 42. This simplifies to 8x ≥ 22. Adding the same value to both sides of an inequality maintains the inequality. This is a fundamental property of inequalities and is crucial for solving them. Isolating the variable term is a key step in solving any equation or inequality. It brings us closer to finding the value or range of values that satisfy the given condition. By isolating 8x, we've set the stage for the final step of dividing by the coefficient of x. The principle behind isolating the variable term is to undo any operations that are being performed on the variable. In this case, we were subtracting 42, so we added 42 to both sides to undo the subtraction. This process is similar to solving equations, where we use inverse operations to isolate the variable. By carefully applying these principles, we can effectively solve inequalities and find the solution set.

Step 4: Solve for x

Now that we have isolated the variable term, 8x ≥ 22, the final step is to solve for x. To do this, we need to divide both sides of the inequality by the coefficient of x, which is 8. This gives us: 8x / 8 ≥ 22 / 8. Simplifying this, we get x ≥ 2.75. Dividing both sides of an inequality by a positive number maintains the direction of the inequality. This is another fundamental property of inequalities. If we were dividing by a negative number, we would need to reverse the inequality sign. Solving for x gives us the solution set, which is the range of values that satisfy the original inequality. In this case, x must be greater than or equal to 2.75. This means that any value of x that is 2.75 or larger will make the inequality true. We can represent this solution set graphically on a number line or in interval notation. Solving for x is the ultimate goal of solving an inequality. It tells us the values that the variable can take to satisfy the given condition. By carefully applying the properties of inequalities, we can isolate the variable and find the solution set.

The Solution Set

The solution set for the inequality 8(x-5)-2 ≥ -20 is x ≥ 2.75. This means that any value of x that is greater than or equal to 2.75 will satisfy the inequality. We can represent this solution set in a few different ways:

  • Graphically: On a number line, we would draw a closed circle (or a square bracket) at 2.75 and shade the line to the right, indicating that all values greater than 2.75 are included in the solution set.
  • Interval Notation: In interval notation, the solution set is written as [2.75, ∞). The square bracket indicates that 2.75 is included in the solution set, and the infinity symbol (∞) indicates that the solution set extends indefinitely to the right.

Understanding the solution set is crucial for interpreting the results of an inequality. It tells us the range of values that the variable can take to make the inequality true. This is particularly important in real-world applications, where the solution set might represent a range of possible outcomes or values. For example, in a budgeting problem, the solution set might represent the range of affordable prices for a particular item. The solution set can also be used to check the solution to an inequality. We can choose a value from the solution set and plug it back into the original inequality to see if it makes the inequality true. If it does, then our solution is likely correct. Conversely, we can choose a value outside the solution set and plug it into the original inequality. If it makes the inequality false, then our solution is also likely correct.

Choosing the Correct Answer

Now, let's revisit the original question and the given answer choices. The question was: What is the solution set of the inequality 8(x-5)-2 ≥ -20? The answer choices were:

  • A. x ≥ 4
  • B. x ≥ -3
  • C. x ≤ 12
  • D. x ≤ -12

We found the solution set to be x ≥ 2.75. Comparing this to the answer choices, we can see that none of the options exactly match our solution. However, option A, x ≥ 4, is the closest. To determine if option A is correct, we need to consider whether 4 is greater than or equal to 2.75, which it is not. If we calculate 22/8, we get 2.75. Thus, the correct answer should be x ≥ 2.75. Since this option is not available, there might be a mistake in the provided options. If we were to approximate 2.75 to the nearest whole number, it would be 3. However, x ≥ 3 is still not in the options. It's important to always double-check your work and the given options to ensure accuracy. If you encounter a situation where none of the options seem correct, it's possible that there's an error in the question or the answer choices. In such cases, it's best to consult with your instructor or a reliable source to clarify the situation.

Conclusion

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. In this article, we've walked through the process of solving the inequality 8(x-5)-2 ≥ -20, breaking down each step and explaining the underlying principles. We started by distributing, then combined like terms, isolated the variable term, and finally solved for x. We also discussed how to represent the solution set graphically and in interval notation. While the answer choices provided didn't perfectly match our solution, we learned the importance of double-checking our work and considering the possibility of errors in the question or options. By mastering the techniques presented in this article, you'll be well-equipped to tackle a wide variety of inequality problems. Remember to practice regularly and seek clarification when needed. With a solid understanding of inequalities, you'll be able to apply these concepts to solve real-world problems and excel in your mathematical studies. The key to success in solving inequalities, and indeed in all areas of mathematics, is to understand the fundamental principles and practice applying them consistently. By building a strong foundation, you'll be able to tackle more complex problems with confidence and accuracy.