Solving $\frac{m}{-7} \leq 14$ A Step-by-Step Guide

Hey guys! Today, we're diving into a common question in mathematics: How do we solve inequalities? Specifically, we're tackling the inequality $\frac{m}{-7} \leq 14$. This is a fundamental concept in algebra, and understanding how to solve these types of problems is crucial for more advanced math. Inequalities are mathematical statements that compare two expressions using symbols like 'less than' ($<$), 'greater than' ($>$), 'less than or equal to' ($\leq$), and 'greater than or equal to' ($\geq$). Unlike equations, which have one specific solution, inequalities often have a range of solutions. Our goal here is to isolate the variable, 'm' in this case, to determine the range of values that satisfy the inequality. So, let's break down the steps and understand the key principles involved. Before we jump into the specific problem, it’s important to remember a golden rule when dealing with inequalities: 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 because multiplying or dividing by a negative number changes the sign of the values, and to maintain the truth of the inequality, we need to reverse the comparison. For instance, if we have -2 < 4, multiplying both sides by -1 gives us 2 > -4, where the '<' sign has been flipped to '>'. Keeping this rule in mind will help us avoid common mistakes and arrive at the correct solution. Now, let’s get started with our problem!

Understanding the Inequality $\frac{m}{-7} \leq 14$

Our main goal here is to isolate the variable m on one side of the inequality. The inequality we're working with is $\frac{m}{-7} \leq 14$. To get m by itself, we need to undo the operation that's being applied to it. In this case, m is being divided by -7. The inverse operation of division is multiplication, so we'll need to multiply both sides of the inequality by -7. But remember the golden rule we talked about earlier! Since we're multiplying by a negative number, we need to flip the inequality sign. This is a critical step, and forgetting to do this will lead to an incorrect answer. So, let's visualize what happens when we apply this operation. We start with $\frac{m}{-7} \leq 14$. Multiplying both sides by -7, we get $(-7) \cdot \frac{m}{-7} \geq 14 \cdot (-7)$. Notice how the '$\leq$' sign has been flipped to '$\geq$'. This is because we're multiplying by a negative number. On the left side, the -7 in the numerator and the -7 in the denominator cancel each other out, leaving us with just m. On the right side, 14 multiplied by -7 gives us -98. So, our inequality now looks like this: $m \geq -98$. This tells us that m is greater than or equal to -98. Any value of m that is -98 or larger will satisfy the original inequality. Think about it this way: if we were to substitute -98 for m in the original inequality, we'd get $\frac{-98}{-7} \leq 14$, which simplifies to 14 $\leq$ 14, a true statement. If we chose a value larger than -98, like 0, we'd get $\frac{0}{-7} \leq 14$, which simplifies to 0 $\leq$ 14, also a true statement. But if we chose a value smaller than -98, like -100, we'd get $\frac{-100}{-7} \leq 14$, which simplifies to approximately 14.29 $\leq$ 14, a false statement. This confirms that our solution, $m \geq -98$, is correct. Now that we understand the mechanics of solving this inequality, let's look at the multiple-choice options provided and see which one correctly describes the necessary operation.

Analyzing the Options

Let's break down each of the multiple-choice options to determine the correct step in solving the inequality $\frac{m}{-7} \leq 14$:

• A. Divide both sides by 14. This option is incorrect. Dividing both sides by 14 wouldn't help us isolate m. It would only complicate the inequality further. We need to undo the division by -7, and dividing by 14 doesn't achieve that. Think of it like trying to untie a knot by pulling on the wrong string; it just makes things tighter. To effectively isolate m, we need to perform the inverse operation of dividing by -7, which is multiplying by -7. Dividing by 14 doesn't address the -7 in the denominator, so it's not the right approach.

• B. Divide both sides by -7. This option is also incorrect. While dividing by -7 might seem like the right approach since we want to get rid of the -7 in the denominator, it actually reverses the original operation. We want to undo the division by -7, not repeat it. Moreover, if we were to divide both sides by -7, we'd end up with a more complex expression on the left side and a different constant on the right side, moving us further away from isolating m. The key is to perform the inverse operation, which in this case is multiplication.

• C. Multiply both sides by 14. This option is incorrect as well. Multiplying by 14 is an arbitrary operation that doesn't help us isolate m. It's like adding a random ingredient to a recipe; it might not ruin the dish, but it certainly won't improve it. To solve for m, we need to focus on the existing operation being applied to m, which is division by -7. Multiplying by 14 doesn't address this operation, so it's not the correct step.

• D. Multiply both sides by -7. This is the correct option! As we discussed earlier, multiplying both sides by -7 is the key to isolating m. It undoes the division by -7, allowing us to get m by itself. And importantly, it reminds us of the crucial rule of flipping the inequality sign when multiplying by a negative number. This option directly addresses the operation being applied to m and provides the necessary step to solve the inequality. By multiplying both sides by -7, we effectively cancel out the -7 in the denominator and move closer to our solution.

Therefore, the correct answer is D. Multiply both sides by -7. This is the step that directly addresses the operation being applied to m and allows us to isolate the variable. Remember, when dealing with inequalities, it's crucial to perform the inverse operation and pay attention to the sign of the number you're multiplying or dividing by.

The Importance of Flipping the Inequality Sign

Let's take a moment to really emphasize why flipping the inequality sign when multiplying or dividing by a negative number is so important. This is a concept that often trips students up, but understanding the reasoning behind it can make all the difference. Imagine a simple inequality: -2 < 4. This statement is clearly true. -2 is indeed less than 4. Now, let's say we forget the rule and multiply both sides by -1 without flipping the sign. We'd get 2 < -4. This statement is completely false! 2 is not less than -4. This simple example illustrates why flipping the inequality sign is necessary. Multiplying or dividing by a negative number essentially reverses the number line. What was on the left becomes the right, and vice versa. So, to maintain the truth of the inequality, we need to reverse the comparison as well. Think of it like looking in a mirror. Your left hand appears on the right, and your right hand appears on the left. The image is flipped. Similarly, multiplying or dividing by a negative number flips the relationship between the two sides of the inequality. To put it another way, multiplying by a negative number changes the direction of the inequality. If something was less than, it becomes greater than, and vice versa. If we didn't flip the sign, we'd be making a false statement. In our original problem, $\frac{m}{-7} \leq 14$, multiplying both sides by -7 requires us to flip the '$\leq$' sign to '$\geq$'. If we didn't, we'd end up with an incorrect solution set. The solution $m \geq -98$ includes all values of m that are greater than or equal to -98. If we didn't flip the sign, we'd get $m \leq -98$, which would include values less than or equal to -98. These are two completely different sets of numbers! So, remember the golden rule: Always flip the inequality sign when multiplying or dividing by a negative number. This is a crucial step in solving inequalities correctly. By understanding the reasoning behind this rule, you can avoid common mistakes and confidently solve a wide range of inequality problems. Now that we've reinforced this key concept, let's move on to a quick recap of the steps we've taken to solve our inequality.

Recap and Final Thoughts

Alright guys, let's do a quick recap of what we've covered in solving the inequality $\frac{m}{-7} \leq 14$. This will help solidify the process and ensure you're comfortable tackling similar problems in the future. First, we identified the goal: to isolate the variable m. This is the fundamental principle behind solving any equation or inequality. We want to get m by itself on one side of the inequality so we can determine the range of values that satisfy the condition. Next, we recognized that m was being divided by -7. To undo this operation, we needed to perform the inverse operation, which is multiplication. So, we multiplied both sides of the inequality by -7. And here's the critical part: We remembered the golden rule! Because we were multiplying by a negative number, we had to flip the inequality sign. This is the step that often gets overlooked, but it's absolutely essential for arriving at the correct solution. By multiplying both sides by -7 and flipping the sign, we transformed the inequality into $m \geq -98$. This is our solution! It tells us that any value of m that is greater than or equal to -98 will satisfy the original inequality. We then analyzed the multiple-choice options and determined that option D, "Multiply both sides by -7," was the correct answer. We ruled out the other options because they didn't address the operation being applied to m or they performed the operation incorrectly. Finally, we emphasized the importance of flipping the inequality sign when multiplying or dividing by a negative number. We discussed the reasoning behind this rule and showed how failing to flip the sign can lead to an incorrect solution. Inequalities are a fundamental concept in mathematics, and mastering them is crucial for success in algebra and beyond. By understanding the principles involved and practicing regularly, you can become confident in your ability to solve a wide range of inequality problems. Remember to always focus on isolating the variable, perform the inverse operations, and pay close attention to the sign of the numbers you're working with. With these skills, you'll be well-equipped to tackle any inequality that comes your way! So keep practicing, stay curious, and you'll become a math whiz in no time!