Solving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of inequalities and figure out how to solve them. Specifically, we're going to tackle the inequality $-\frac{2}{7} v-6<-18$. Don't worry, it might look a bit intimidating at first, but trust me, it's totally manageable. We'll break it down step by step, making sure you understand every move. Our goal? To isolate 'v' and find the range of values that make the inequality true. Ready to roll up our sleeves and get started? Let's do it!

Understanding the Basics of Inequalities

Before we jump into solving the specific inequality, let's quickly review what inequalities are all about. Think of an inequality as a mathematical statement that compares two values, but instead of saying they're equal (like in an equation), it says one is greater than, less than, greater than or equal to, or less than or equal to the other. We use symbols like '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to) to represent these relationships. Unlike equations, which usually have a single solution, inequalities often have a range of solutions. This means there's a whole set of numbers that satisfy the condition. The core principle for solving inequalities is pretty similar to solving equations: we want to isolate the variable (in our case, 'v') on one side of the inequality. However, there's one important rule to remember: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. This is a crucial step that many people overlook, so pay close attention! Also, the concepts we'll use include addition, subtraction, multiplication, and division properties of inequalities. These properties allow us to manipulate the inequality while preserving its truth. These properties guarantee that the manipulations we perform will not change the solution set of the inequality. Also, understanding the properties of inequalities will help you solve problems. Keep in mind that solving inequalities can be applied to real-world scenarios. For example, you might use inequalities to represent budget constraints, age restrictions, or performance requirements.

Step 1: Isolating the Variable - Get Started

Alright, let's start solving our inequality: $-\frac{2}{7} v-6<-18$. The first thing we want to do is isolate the term containing 'v'. To do this, we need to get rid of that '-6' that's hanging out on the same side as 'v'. We can do this by using the addition property of inequality. Add 6 to both sides of the inequality. This will cancel out the '-6' on the left side. Remember, whatever you do to one side of the inequality, you must do to the other side to keep things balanced. So, we'll have:

-\frac{2}{7} v-6 + 6 < -18 + 6

This simplifies to:

-\frac{2}{7} v < -12

See? We're already making progress! By adding 6 to both sides, we've simplified the inequality and gotten closer to isolating 'v'. This step helps to ensure that we maintain the balance of the inequality. Also, this stage requires a strong understanding of how to use the addition property of inequality. The addition property of inequality is fundamental in solving this type of problem. Without a solid understanding of this, it can be difficult to proceed. It's like having the building blocks for constructing a house. You must know how to place each one to complete the structure properly. Don't worry if it takes a little practice to get comfortable with the properties. With a little practice, you'll be solving inequalities like a pro!

Step 2: Isolating the Variable - Multiplying with Caution

Now that we have $-\frac{2}{7} v < -12$, we need to get rid of the coefficient $-\frac{2}{7}$ that's multiplying 'v'. We can do this by multiplying both sides of the inequality by the reciprocal of $-\frac{2}{7}$, which is $-\frac{7}{2}$. But, here's the crucial part: Since we're multiplying by a negative number, we must flip the direction of the inequality sign. Remember what we talked about earlier? This is where that rule comes into play! So, we'll have:

(-\frac{7}{2}) * (-\frac{2}{7} v) > (-12) * (-\frac{7}{2})

Notice how the '<' sign has now become '>'. This is because we multiplied by a negative number. This step is a critical aspect of solving inequalities and the most common place where mistakes are made. It's really important to keep track of that sign change. When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed to preserve the validity of the inequality. Doing this correctly ensures that our final solution set accurately reflects the possible values of 'v'. This is a very important step and the core of the problem. If you forget to flip the inequality sign, your solution will be incorrect. This step might seem confusing at first, but with a little practice, you'll be able to master it easily. This step requires a thorough understanding of the multiplication property of inequality. Ensure that you have a firm grasp of this concept. Don't rush; take your time. This careful approach can help you avoid making simple errors. And also, you must remember, when multiplying or dividing both sides by a negative number, the inequality sign flips.

Step 3: Simplifying and Finding the Solution

Okay, let's simplify the expression we got in the previous step, $(-\frac{7}{2}) * (-\frac{2}{7} v) > (-12) * (-\frac{7}{2})$. The left side simplifies to 'v' (the negative signs cancel out, and the fractions also simplify), and the right side simplifies to 42 (because $-12 * -7/2 = 84/2 = 42$). So, we're left with:

v > 42

And there you have it! We've solved the inequality. This means that any value of 'v' that is greater than 42 will satisfy the original inequality. The solution is v > 42. This is the final step, and it is a straightforward process. The key is to remember the rules we discussed earlier. Ensure that you have performed all previous steps correctly, and simplify to get the solution. Now that we have the answer, we can check it. Remember, you can always test your solution by plugging in a value greater than 42 into the original inequality. If you substitute a value, for example, 43, the inequality should hold true. This will give you confidence that you have solved the problem correctly. This step is also an important part of the solution. If the answer is not correct, you can go back and check each step. This process helps solidify your understanding of solving inequalities. You can also represent the solution on a number line, which can help visualize the solution set. This will show you all the values of 'v' that make the inequality true. You're now equipped to solve inequalities! This is a great achievement. Congrats!

Visualizing the Solution: Number Line

Let's visualize the solution, v > 42, on a number line. This is a great way to understand the solution set. Draw a number line. Mark a point at 42. Since 'v' is greater than 42 (and not equal to), we use an open circle at 42. This indicates that 42 itself is not included in the solution. Then, draw an arrow pointing to the right from 42. This arrow represents all the numbers greater than 42. Every point on the arrow is a valid solution. This visualization clearly shows that the solution includes all numbers larger than 42, but excludes 42 itself. Visualizing the solution on a number line gives you a clear picture of what the solution represents. It helps to clarify the solution set. This approach simplifies the understanding of the inequality's solution. Also, this approach can help you determine the solution. Remember that an open circle represents that the value is not included, and a closed circle represents the inclusion. The direction of the arrow indicates the range of values that satisfy the inequality.

Checking Your Work: Verification

It's always a good idea to check your solution. Let's pick a number that is greater than 42, say 43. Substitute '43' for 'v' in the original inequality, which is $-\frac{2}{7} v-6<-18$: $-\frac{2}{7}(43) - 6 < -18$. Calculate $-\frac{2}{7}(43)$, which is approximately -12.29. Then, subtract 6: -12.29 - 6 = -18.29. Is -18.29 < -18? Yes, it is! So, the inequality holds true. This confirms that our solution, v > 42, is correct. Checking your work is a critical step in problem-solving. It helps to verify the accuracy of your solution. Substituting a number from the solution set helps to ensure that you have understood the problem correctly. If your check doesn't work, don't worry. This is a chance to review your steps and identify any errors. With practice, you'll become more comfortable with this process. And you will be able to verify your answers. Always check your answers to enhance your understanding. Remember, practice makes perfect! So, keep practicing, and you'll become a pro at solving inequalities. Keep going!

Conclusion: Mastering Inequalities

Alright, guys, you've successfully solved the inequality $-\frac{2}{7} v-6<-18$ and found that v > 42. You've learned the key steps involved: isolating the variable, paying attention to the inequality sign when multiplying or dividing by negative numbers, and visualizing the solution on a number line. Remember that practice is key to mastering inequalities. Try solving more problems on your own. You can find plenty of exercises online or in textbooks. The more you practice, the more confident you'll become. Each inequality you solve will strengthen your skills. And remember to always check your answers to ensure they are correct. Now you're well on your way to conquering more complex mathematical problems. Keep up the excellent work, and always remember to apply what you have learned. Solving inequalities has many real-world applications. Understanding these concepts will help you build a solid foundation. You've got this! Keep practicing, and you'll be a pro in no time! Keep practicing, and you'll be a pro in no time! Keep going; you're doing great!