Solving Inequalities Solution Set For 8(x-5) - 3x ≥ -20

Introduction

In this article, we will delve into the process of solving a linear inequality. Specifically, we'll address the inequality 8(x-5) - 3x ≥ -20 and determine its solution set. Understanding how to solve inequalities is a fundamental skill in algebra, with applications in various fields of mathematics and beyond. This exploration will not only provide the answer but also guide you through the steps involved, ensuring a solid grasp of the underlying concepts. This article aims to provide a comprehensive explanation of solving inequalities, making it accessible to learners of all levels. We will break down each step, clarify the reasoning behind it, and highlight common pitfalls to avoid. Whether you are a student tackling algebra problems or someone looking to refresh your math skills, this guide will equip you with the knowledge and confidence to handle similar problems with ease. By the end of this discussion, you should be able to not only solve this specific inequality but also apply the techniques to solve other linear inequalities. The key to mastering inequalities lies in understanding the rules and applying them systematically. So, let's embark on this journey together and unravel the solution to this inequality, enhancing your algebraic problem-solving abilities along the way. We will also provide additional examples and practice problems to solidify your understanding and ensure you are well-prepared for any inequality-related challenges you may encounter. Remember, mathematics is a building-block subject, and a strong foundation in algebra is crucial for success in more advanced topics.

Step-by-Step Solution

To solve the inequality 8(x-5) - 3x ≥ -20, we'll follow a systematic approach involving simplification, combining like terms, isolating the variable, and expressing the solution set. Our primary goal is to isolate x on one side of the inequality. This involves performing operations that maintain the truth of the inequality while gradually simplifying it. Remember, when multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality sign. This is a crucial rule to remember, as it directly impacts the accuracy of your solution. Neglecting to flip the sign when necessary is a common mistake that can lead to an incorrect solution set. Throughout this process, we will emphasize the importance of each step and the reasoning behind it, providing a clear and concise explanation. This will not only help you solve this specific problem but also equip you with the skills to tackle a wide range of inequality problems. We will also highlight the parallels between solving inequalities and solving equations, emphasizing the key differences that must be considered. Understanding these nuances is essential for mastering algebraic problem-solving. Furthermore, we will discuss how to represent the solution set graphically on a number line, providing a visual representation of the range of values that satisfy the inequality. This visual aid can be particularly helpful for understanding the solution set and communicating it effectively.

1. Distribute the 8

Our first step involves distributing the 8 across the terms inside the parentheses. This is a crucial step in simplifying the expression and allows us to eliminate the parentheses, making it easier to combine like terms. The distributive property states that a(b + c) = ab + ac. Applying this property correctly is fundamental to solving algebraic equations and inequalities. In this case, we multiply 8 by both x and -5. This process transforms the expression 8(x - 5) into 8x - 40. Ensuring accuracy in this step is vital, as any error here will propagate through the rest of the solution. It's a good practice to double-check your work, especially when dealing with multiple terms and operations. This simple check can prevent costly mistakes and save time in the long run. Moreover, understanding the distributive property is essential for more advanced algebraic manipulations, such as factoring and expanding polynomials. Therefore, mastering this concept is a worthwhile investment in your mathematical skills. In the context of real-world problems, the distributive property is used in various applications, such as calculating costs, determining areas, and modeling physical phenomena. For example, if you are buying multiple items at a store with a discount applied to the total, you would use the distributive property to calculate the final price. This practical relevance underscores the importance of mastering this algebraic concept.

So, the inequality becomes:

8x - 40 - 3x ≥ -20

2. Combine Like Terms

Next, we combine the like terms on the left side of the inequality. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with x: 8x and -3x. Combining these terms simplifies the inequality and brings us closer to isolating x. To combine like terms, we simply add or subtract their coefficients. The coefficient is the numerical factor that multiplies the variable. In our example, the coefficient of 8x is 8, and the coefficient of -3x is -3. Adding these coefficients, we get 8 - 3 = 5. Therefore, combining 8x and -3x results in 5x. This simplification is a key step in solving the inequality, making it easier to isolate the variable and determine the solution set. It is essential to pay close attention to the signs of the coefficients when combining like terms. A mistake in the sign can lead to an incorrect result. It's also important to remember that only like terms can be combined. Terms with different variables or different powers of the same variable cannot be combined. For example, 5x and 5x² are not like terms and cannot be combined. Understanding the concept of like terms is fundamental to algebraic manipulation and simplification. It is used extensively in solving equations, inequalities, and other algebraic problems. This skill is also applicable in various real-world scenarios, such as simplifying expressions in physics, engineering, and economics. For instance, when calculating the total cost of multiple items with different prices, you would combine the prices of similar items to simplify the calculation. Therefore, mastering the ability to combine like terms is a valuable asset in both mathematics and practical applications.

So, the inequality now looks like this:

5x - 40 ≥ -20

3. Isolate the Variable Term

To further isolate x, we need to get rid of the constant term, -40, on the left side of the inequality. We achieve this by adding 40 to both sides of the inequality. Adding the same value to both sides maintains the balance of the inequality, ensuring that the solution set remains unchanged. This is a fundamental principle in solving both equations and inequalities. The goal of this step is to isolate the term containing the variable x on one side of the inequality. In this case, we want to isolate 5x. Adding 40 to both sides cancels out the -40 on the left side, leaving us with just 5x on that side. On the right side, we add 40 to -20, which results in 20. It is crucial to perform the same operation on both sides of the inequality to maintain its validity. Failing to do so will lead to an incorrect solution. This principle applies to all operations, including addition, subtraction, multiplication, and division. When solving more complex inequalities, this step may involve multiple additions or subtractions to isolate the variable term. The key is to systematically eliminate any constant terms on the same side as the variable term. Understanding this process is essential for solving a wide range of algebraic problems. It is also applicable in various real-world scenarios, such as solving for an unknown quantity in a budget calculation or determining the required input to achieve a desired output in a manufacturing process. Therefore, mastering the technique of isolating the variable term is a valuable skill in both mathematics and practical applications.

Adding 40 to both sides, we get:

5x - 40 + 40 ≥ -20 + 40

5x ≥ 20

4. Solve for x

Finally, to solve for x, we divide both sides of the inequality by 5. This isolates x and gives us the solution set. Division, like addition, subtraction, and multiplication, must be performed on both sides of the inequality to maintain its balance. In this case, dividing both sides by 5 will isolate x on the left side. It is crucial to remember the rule about flipping the inequality sign when dividing by a negative number. However, since we are dividing by a positive number (5), the inequality sign remains the same. This is a critical distinction that must be carefully considered when solving inequalities. Dividing 5x by 5 gives us x, and dividing 20 by 5 gives us 4. Therefore, the solution to the inequality is x ≥ 4. This means that any value of x that is greater than or equal to 4 will satisfy the original inequality. This solution set can be represented graphically on a number line, where we would shade the region to the right of 4, including 4 itself. Understanding how to solve for x in inequalities is a fundamental skill in algebra. It is used extensively in solving various types of problems, including optimization problems, constraint satisfaction problems, and modeling real-world scenarios. For instance, you might use inequalities to determine the range of values that satisfy a budget constraint or to find the optimal production level that maximizes profit. Therefore, mastering this skill is a valuable asset in both mathematics and practical applications.

Dividing both sides by 5, we have:

5x / 5 ≥ 20 / 5

x ≥ 4

Solution Set

The solution set for the inequality 8(x-5) - 3x ≥ -20 is x ≥ 4. This means that any value of x greater than or equal to 4 will satisfy the inequality. The solution set represents a range of values, rather than a single value as in an equation. This is a key difference between solving inequalities and solving equations. Understanding the solution set is crucial for interpreting the results and applying them in real-world contexts. For example, if the inequality represents a constraint on a certain variable in a problem, the solution set tells us the range of values that satisfy that constraint. The solution set can be represented in various ways, including algebraically (as we have done here), graphically on a number line, and in interval notation. Each representation provides a different perspective on the solution and can be useful in different situations. The algebraic representation, x ≥ 4, is the most concise and precise way to express the solution set. The graphical representation provides a visual understanding of the solution set, showing the range of values that satisfy the inequality. Interval notation is a more compact way of representing the solution set, especially for more complex inequalities. In this case, the solution set x ≥ 4 can be written in interval notation as [4, ∞). Understanding these different representations is essential for effectively communicating and interpreting solutions to inequalities. The concept of a solution set is fundamental to mathematics and is used extensively in various fields, including algebra, calculus, and real analysis. It is also applicable in various real-world scenarios, such as optimization problems, decision-making, and risk assessment. Therefore, mastering the concept of a solution set is a valuable asset in both mathematics and practical applications.

Answer

The correct answer is A. x ≥ 4.

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Solving Inequalities Find the Solution Set for 8(x-5) - 3x ≥ -20