Solving Inequalities A Step-by-Step Guide To -4t ≤ 48

Jul 24, 2025 by ADMIN 54 views

$Solving Inequalities A Step-by-Step Guide to $-4t \leq 48$$

Hey guys! Let's dive into solving the inequality $-4t leq 48$ . Inequalities might seem a bit tricky at first, but with a clear step-by-step approach, you'll find them quite manageable. This article will guide you through the process, ensuring you understand each step along the way. We'll break down the problem, discuss the key concepts, and provide a comprehensive solution. So, grab your pencils, and let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are. Unlike equations that state two expressions are equal, inequalities compare expressions using symbols like less than ( $<$ ), greater than ( $>$ ), less than or equal to ( $leq$ ), and greater than or equal to ( $leq$ ). Understanding these symbols is crucial for solving and interpreting inequalities correctly. For instance, $-4t leq 48$ means that the value of $-4t$ is less than or equal to 48. Inequalities are fundamental in mathematics, showing up in various fields such as algebra, calculus, and real analysis. They help us define ranges and constraints, which are essential in problem-solving and mathematical modeling. Recognizing the type of inequality you're dealing with—whether it's linear, quadratic, or another form—is the first step in choosing the appropriate solution method. Linear inequalities, like the one we're tackling today, involve variables raised to the power of one and can be solved using techniques similar to those used for linear equations. However, there's a crucial difference to keep in mind: multiplying or dividing by a negative number flips the inequality sign. This is a key concept we'll explore further as we solve our example. Beyond linear inequalities, quadratic inequalities involve variables raised to the power of two, and their solutions often require factoring or using the quadratic formula. Other types of inequalities, such as polynomial and rational inequalities, may require additional techniques like sign analysis. No matter the type, understanding the core principles of inequalities is essential for success in mathematics. So, as we work through our example, keep in mind the basic rules and how they apply to each step. Mastering inequalities not only helps in solving mathematical problems but also enhances logical thinking and problem-solving skills, which are valuable in many areas of life.

The Key Principle: Flipping the Inequality Sign

One of the most important things to remember when working with inequalities is that if you multiply or divide both sides by a negative number, you need to flip the inequality sign. This might seem a bit counterintuitive at first, but it's essential for maintaining the truth of the inequality. To illustrate why this is necessary, let's consider a simple example. Suppose we have the inequality $2 < 4$ . This statement is clearly true. Now, if we multiply both sides by -1 without flipping the sign, we get $-2 < -4$ , which is false. However, if we flip the sign, we get $-2 > -4$ , which is the correct relationship. Flipping the inequality sign ensures that the inequality remains accurate after multiplying or dividing by a negative number. The reason behind this rule lies in the number line. When you multiply a positive number by a negative number, its position on the number line is reflected across zero. This reflection changes the order of numbers; what was smaller becomes larger, and vice versa. This principle applies not just to simple inequalities but to more complex ones as well. When solving inequalities involving variables, you must be mindful of this rule whenever you multiply or divide by a negative number. Forgetting to flip the sign can lead to an incorrect solution, so it's a critical step to remember. This concept is particularly important when dealing with linear inequalities, where you might need to divide by a negative coefficient to isolate the variable. In our problem, $-4t leq 48$ , we will encounter this situation when we divide by -4. Understanding this rule thoroughly will help you avoid common mistakes and confidently solve a wide range of inequality problems. It’s not just about memorizing a rule; it’s about grasping the underlying mathematical principle that ensures the correctness of your solution. So, as we move forward with solving our inequality, keep this key concept in mind, and you'll be well on your way to mastering inequalities.

Step-by-Step Solution for $-4t leq 48$

Now, let's get down to business and solve the inequality $-4t leq 48$ . We'll take it one step at a time to make sure everything is clear. Our goal here is to isolate $t$ on one side of the inequality. To do this, we need to undo the multiplication by -4. The inverse operation of multiplication is division, so we'll divide both sides of the inequality by -4. Here’s where that crucial rule about flipping the inequality sign comes into play. Since we're dividing by a negative number, we must flip the $leq$ sign to $leq$ . Let's see how this looks:

-4t leq 48

Divide both sides by -4:

rac{-4t}{-4} leq rac{48}{-4}

Remember to flip the inequality sign:

t leq -12

And there you have it! The solution to the inequality $-4t leq 48$ is $t leq -12$ . This means that any value of $t$ that is greater than or equal to -12 will satisfy the original inequality. Solving inequalities requires careful attention to detail, particularly when dealing with negative numbers. This step-by-step approach ensures we follow all the rules correctly and arrive at the accurate solution. It’s important to show each step clearly, so you can track your progress and identify any potential errors. Dividing both sides by -4 and flipping the sign is the key move in this problem. This demonstrates a fundamental principle in solving inequalities, and it’s a technique that will be used repeatedly in more complex problems. Practicing these steps will make you more comfortable with the process and help you build confidence in your ability to solve inequalities. So, let’s recap: we identified the operation needed to isolate the variable, divided both sides by the coefficient of the variable (a negative number), and most importantly, remembered to flip the inequality sign. By following these steps, we successfully solved the inequality and found the range of values that satisfy the condition.

Expressing the Solution

Now that we've found the solution $t leq -12$ , let's talk about how to express it in different ways. Understanding how to represent solutions is just as important as finding them. There are a few common methods we can use: inequality notation, number line representation, and interval notation. We've already seen the solution in inequality notation: $t leq -12$ . This is a concise way to state that $t$ can be any number greater than or equal to -12. But let's explore the other methods. Expressing solutions effectively helps in visualizing and understanding the range of values that satisfy the inequality. A number line representation is a visual way to show the solution set. To represent $t leq -12$ on a number line, draw a number line and locate -12. Since $t$ can be equal to -12, we use a closed circle (or a bracket) at -12. Then, we shade the line to the right of -12 to indicate that all values greater than -12 are also part of the solution. This visual representation makes it easy to see the range of possible values for $t$ . Interval notation is another way to express the solution set. It uses intervals and parentheses or brackets to indicate the range of values. For $t leq -12$ , the interval notation is $[-12, infty)$ . The bracket on the -12 indicates that -12 is included in the solution, and the parenthesis on the infinity symbol indicates that infinity is not a specific number but rather an unbounded range. Using interval notation provides a clear and standardized way to communicate solution sets. Each of these methods—inequality notation, number line representation, and interval notation—offers a different perspective on the solution. Being comfortable with all three methods allows you to choose the most appropriate one for a given situation and ensures clear communication of your results. So, let's remember that while finding the solution is crucial, knowing how to express it effectively is equally important. By mastering these different representations, you enhance your understanding of inequalities and your ability to work with them in various contexts.

Common Mistakes to Avoid

When solving inequalities, there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you get the correct solution. One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing by a negative number. As we've emphasized, this is a crucial step, and overlooking it will lead to an incorrect answer. Another common mistake is misinterpreting the inequality symbols. Make sure you understand the difference between $<$ , $>$ , $leq$ , and $leq$ . Each symbol has a specific meaning, and using the wrong one can change the solution set. Avoiding common mistakes is essential for accuracy when solving inequalities. For instance, confusing