Solving The Inequality M/-7 ≤ 14 A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of inequalities and tackling a problem that might seem a bit tricky at first glance. But don't worry, we'll break it down step by step and make sure you understand exactly how to find the solution. So, let's jump right into it!

Understanding the Problem: The Inequality m/-7 ≤ 14

At the heart of our discussion lies the inequality m/-7 ≤ 14. Now, what does this actually mean? In simple terms, we're looking for all the possible values of 'm' that, when divided by -7, result in a value less than or equal to 14. Inequalities like this are common in mathematics and have practical applications in various fields, from economics to engineering. Understanding how to solve them is a crucial skill for any math student.

Before we dive into the solution, let's quickly recap what inequalities are. Unlike equations, which have a single solution or a set of specific solutions, inequalities represent a range of values. The symbols used in inequalities, such as '≤' (less than or equal to), '≥' (greater than or equal to), '<' (less than), and '>' (greater than), help us define these ranges. When solving inequalities, we aim to isolate the variable (in our case, 'm') on one side, just like we do with equations. However, there's a crucial twist when dealing with negative numbers, which we'll explore shortly.

Now, let's talk about the key to cracking this inequality: the multiplication property of inequality. This property states that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is super important, guys, and it's where many students often make mistakes. So, keep this in mind as we move forward. In our case, we're dealing with a division by -7, which means we'll need to multiply by -7 to isolate 'm'. And guess what? That means flipping the inequality sign!

Think of it like this: imagine you have two numbers, say 2 and 3. We know that 2 < 3. Now, if we multiply both sides by -1, we get -2 and -3. But -2 is actually greater than -3. So, the inequality sign flips to -2 > -3. This principle applies to all inequalities when multiplying or dividing by a negative number. Getting this concept down pat is essential for accurately solving inequalities and avoiding common pitfalls.

Step-by-Step Solution: Isolating 'm'

Okay, let's get our hands dirty and solve this inequality step by step. Remember our goal: to isolate 'm' on one side of the inequality. We'll do this by carefully applying the rules of algebra, keeping that crucial rule about flipping the inequality sign in mind.

1. Start with the inequality: m/-7 ≤ 14. This is the problem we're tackling.
2. Multiply both sides by -7: To get 'm' by itself, we need to get rid of the -7 in the denominator. So, we multiply both sides of the inequality by -7. This gives us (m/-7) * (-7) and 14 * (-7). But remember, we're multiplying by a negative number, so we need to flip the inequality sign!
3. Simplify: On the left side, the -7s cancel out, leaving us with 'm'. On the right side, 14 * (-7) equals -98. And don't forget, the inequality sign has flipped from ≤ to ≥.
4. The solution: This leaves us with our solution: m ≥ -98. This means that any value of 'm' that is greater than or equal to -98 will satisfy the original inequality.

So, there you have it! We've successfully solved the inequality. But let's make sure we really understand what this solution means and how to interpret it.

Interpreting the Solution: m ≥ -98

Our solution, m ≥ -98, tells us that 'm' can be any number that is greater than or equal to -98. This includes numbers like -98, -97, -96, -50, 0, 1, 10, 100, and so on. Basically, anything from -98 upwards will work in our original inequality.

Understanding the solution set is just as important as finding the solution itself. Think of it as a range of possibilities. In this case, we have a lower limit of -98, and 'm' can be any value above that. This is a fundamental concept in understanding inequalities. To solidify this understanding, let's explore a couple of examples to see how this solution works in practice.

Let's try a number within our solution set, say m = -90. Plugging this into our original inequality, we get -90/-7 ≤ 14. This simplifies to approximately 12.86 ≤ 14, which is true. So, -90 is indeed a valid solution. Now, let's try a number outside our solution set, say m = -100. Plugging this in, we get -100/-7 ≤ 14, which simplifies to approximately 14.29 ≤ 14. This is false! This confirms that -100 is not a solution, reinforcing our understanding of the solution set.

Visualizing the solution set on a number line can also be incredibly helpful. Imagine a number line stretching from negative infinity to positive infinity. Our solution, m ≥ -98, would be represented by a closed circle (or a bracket) at -98, indicating that -98 is included in the solution, and a line extending to the right, indicating all values greater than -98. This visual representation makes it easy to grasp the range of possible values for 'm'.

Identifying the Correct Answer: Option B

Now that we've diligently solved the inequality and understand what our solution means, let's circle back to the answer choices provided. We need to identify the option that matches our solution, m ≥ -98. A quick glance at the options reveals that option B, m ≥ -98, is the correct answer. Hooray! We've nailed it.

It's always a good practice to double-check your answer, especially in math problems. We've already verified our solution by plugging in values within and outside the solution set. This gives us extra confidence that our answer is indeed correct. In test-taking scenarios, time management is key, but taking a few extra moments to verify your answer can prevent careless mistakes and boost your overall score.

Why Other Options Are Incorrect: A Quick Review

To truly master this type of problem, it's beneficial to understand why the other answer options are incorrect. This helps solidify your understanding of the concepts and prevent similar errors in the future. Let's quickly examine why options A, C, and D are not the correct solutions.

• Option A: m ≥ -2. This is incorrect because it represents a completely different solution set. If we were to plug in a value that satisfies this inequality, like m = 0, into our original inequality, we'd get 0/-7 ≤ 14, which is true. However, this doesn't mean it's the only solution. Our solution set is much broader, including all values greater than or equal to -98.
• Option C: m ≤ -2. This is also incorrect. This inequality suggests that 'm' can be any value less than or equal to -2. If we were to test a value like m = -100, we'd get -100/-7 ≤ 14, which we already know is false. This highlights that option C doesn't align with our derived solution.
• Option D: m ≤ -98. This is the closest to the correct answer but is still incorrect due to the direction of the inequality. This option suggests that 'm' can be any value less than or equal to -98. This would exclude values like -97, -96, etc., which we know are valid solutions. The critical error here is not flipping the inequality sign when multiplying by a negative number.

By understanding why these options are wrong, we reinforce our understanding of the correct solution and the process of solving inequalities.

Key Takeaways: Mastering Inequalities

Alright, guys, we've reached the end of our journey into solving this inequality. Let's recap the key takeaways to solidify our understanding and equip ourselves for future challenges.

First and foremost, remember the golden rule: when multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign! This is the most common mistake students make, so keep this etched in your mind. Understanding the why behind this rule is crucial. Think about how negative numbers work on the number line, and you'll always remember to flip that sign!

Secondly, practice makes perfect! The more you work with inequalities, the more comfortable you'll become with the process. Try different examples, challenge yourself with more complex problems, and don't be afraid to make mistakes. Mistakes are learning opportunities! The goal here is not to be perfect but to improve and grow in your understanding of the topic.

Thirdly, always interpret your solution. Don't just stop at isolating the variable. Understand what the solution set means and how it relates to the original inequality. Plugging in values to test your solution is a great way to verify your answer and deepen your understanding. It's like a built-in safety check for your work.

Lastly, visualize the solution set. Using a number line to represent the solution can make it much easier to grasp the range of possible values. This visual representation can be particularly helpful when dealing with more complex inequalities or when comparing different solution sets.

By mastering these key takeaways, you'll be well on your way to conquering any inequality that comes your way. Keep practicing, keep exploring, and keep that math brain sharp!

Wrapping Up: Feeling Confident with Inequalities

Well, there you have it! We've successfully navigated the world of inequalities and found the solution to our problem: m/-7 ≤ 14. We've not only found the answer but also delved into the why behind the solution, exploring the key concepts and potential pitfalls along the way. You guys should now feel confident in your ability to tackle similar problems. Solving inequalities, like any mathematical skill, becomes easier and more intuitive with practice and a solid understanding of the underlying principles.

Remember, mathematics is not just about memorizing formulas and procedures; it's about developing critical thinking skills and problem-solving abilities. By breaking down complex problems into smaller, manageable steps, and by understanding the logic behind each step, you can conquer even the most daunting mathematical challenges. So, keep exploring, keep questioning, and keep learning. The world of mathematics is vast and fascinating, and you've just taken another step on your mathematical journey. Great job, everyone!