Solving Linear Inequalities: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into the world of linear inequalities, specifically tackling the problem: 4(3c + 10) < 12c + 40. Don't worry if inequalities seem a little tricky at first; we'll break down the process step by step, making it easy to understand and conquer. This article is your guide to not only solving the inequality but also grasping the concepts behind it, ensuring you can confidently tackle similar problems in the future. We'll explore the solution, discuss what it means, and offer some helpful tips along the way. Get ready to flex those math muscles and unlock the secrets of inequalities!
Understanding the Basics: Linear Inequalities
Before we jump into the problem, let's quickly review what a linear inequality is. Simply put, it's a mathematical statement that compares two expressions using an inequality symbol, such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations, which have a single solution (or a finite number of solutions), inequalities often have a range of solutions. These solutions represent all the values that make the inequality true. The core concept behind solving linear inequalities revolves around isolating the variable (in this case, 'c') on one side of the inequality symbol. We achieve this by performing operations on both sides of the inequality, much like solving an equation. However, there's one crucial difference to remember: When multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality symbol. This is a common point of confusion, so we'll keep it in mind as we work through our problem. Furthermore, it's essential to recognize the real-world applications of linear inequalities. They are used in various fields, from economics and finance to engineering and computer science. Understanding them can give you a solid foundation for more complex mathematical concepts.
Step-by-Step Solution: Cracking the Inequality
Now, let's get down to the business of solving 4(3c + 10) < 12c + 40. We'll proceed in a logical, step-by-step fashion to ensure clarity and accuracy. Each step builds on the previous one, and by following this method, you can effectively tackle any linear inequality. The key is to stay organized and pay close attention to detail. Let’s get started.
- Distribute: Our first step is to get rid of those parentheses. We'll distribute the 4 across the terms inside the parentheses. This means multiplying 4 by both 3c and 10. So, 4 * 3c = 12c and 4 * 10 = 40. Our inequality now looks like this: 12c + 40 < 12c + 40.
- Simplify: Next, we need to gather all terms with 'c' on one side and the constant terms on the other. In this case, we have 12c on both sides of the inequality. If we subtract 12c from both sides, we get: 12c - 12c + 40 < 12c - 12c + 40. This simplifies to 40 < 40.
- Analyze the Result: At this point, we've arrived at 40 < 40. This statement is not true. 40 is not less than 40; it's equal to 40. This outcome indicates that there is no solution to this inequality. The variable 'c' does not have any value that can make the original inequality true. The inequality holds no real solution. This is different from situations where we get a value for 'c', such as c > 2 or c < 5.
Interpreting the Solution: No Solution Explained
The result 40 < 40 is a bit of a curveball. It tells us that there's no solution to the inequality. What does this mean in practical terms? It means that no matter what value you substitute for 'c' in the original inequality 4(3c + 10) < 12c + 40, the statement will never be true. This might seem strange, but it's a valid outcome. In the context of solving inequalities, there are three possible types of outcomes: a single solution (like in equations), an infinite number of solutions (all real numbers), or no solution at all. This particular inequality falls into the
