Identifying The Inequality Property Used In Solving 5x < 100

Jul 23, 2025 by ADMIN 61 views

Unveiling the Property Behind Inequality Solutions: A Deep Dive

Hey there, math enthusiasts! Today, we're going to dissect a common type of math problem: solving inequalities. Specifically, we're going to zoom in on the properties that make these solutions possible. Let's take a look at the inequality presented and break down the magic behind each step. Our mission? To pinpoint the property that's in action during the second step.

$\begin{array}{c}
5 x-9  \textless  91 \\
5 x  \textless  100 \\
x  \textless  20
\end{array}

So, let's put on our detective hats and uncover the mystery property!

Decoding the Inequality: Step-by-Step

To really understand which property is at play, we need to walk through each step of the solution. This isn't just about getting the right answer; it's about grasping the why behind the math. Understanding the 'why' is what transforms you from a problem-solver to a math whiz!

Step 1: From $5x - 9 < 91$ to $5x < 100$

Alright, let's rewind to the initial inequality: $5x - 9 < 91$ . Our goal here is to isolate the term with 'x' on one side. Think of it like separating the ingredients you need for a recipe. To do this, we need to get rid of that '-9' hanging out on the left side. How do we do it? We use the addition property of inequality. This property is super handy because it says we can add the same number to both sides of an inequality without messing up the relationship. It's like adding the same amount of water to two different glasses; the water levels might rise, but the difference between them stays the same.

So, we add 9 to both sides:

$5x - 9 + 9 < 91 + 9$

This simplifies to:

$5x < 100$

Voila! We've successfully isolated the '5x' term. We used the addition property of inequality to make this happen, ensuring we kept the inequality balanced and true. The addition property of inequality is a foundational concept in algebra, allowing us to manipulate inequalities while preserving their validity. It's essential for solving a wide range of problems, from simple inequalities like this one to more complex systems of inequalities. Remember, the key is to add the same value to both sides, maintaining the balance and ensuring the solution remains accurate. This property is not just a tool for solving equations; it's a fundamental principle that underpins much of mathematical reasoning. By understanding and applying the addition property correctly, you can confidently tackle inequality problems and build a stronger foundation in algebra. This step highlights the importance of careful manipulation and the power of mathematical properties to simplify complex expressions.

Step 2: From $5x < 100$ to $x < 20$

Now, we arrive at the heart of our quest: the second step. We're starting with $5x < 100$ , and our mission is to isolate 'x' completely. It's like we've got a package labeled '5x,' and we need to unwrap it to reveal just 'x.' What mathematical tool will help us do this? Well, notice that 'x' is being multiplied by 5. To undo this multiplication, we need to do the opposite operation: division. The multiplication property of inequality is our go-to property here, but with a slight twist. It's crucial to remember that when we multiply or divide both sides of an inequality by a negative number, we have to flip the inequality sign. However, in this case, we're dividing by a positive number (5), so we don't need to worry about flipping the sign. It's like driving on a straight road; we can move forward without changing direction.

So, we divide both sides by 5:

$\frac{5x}{5} < \frac{100}{5}$

This simplifies to:

$x < 20$

And there we have it! We've successfully isolated 'x,' revealing that x is less than 20. The multiplication property of inequality is the hero of this step. It allowed us to divide both sides by 5, maintaining the balance of the inequality and leading us to our solution. This property is a cornerstone of inequality manipulation, enabling us to solve a vast array of problems. It's not just about performing the division; it's about understanding the underlying principle of maintaining the relationship between the two sides of the inequality. By dividing both sides by a positive number, we ensure that the inequality remains valid and the solution set is accurately determined. This step emphasizes the importance of recognizing the operations involved and applying the appropriate properties to achieve the desired result. The multiplication property, when applied correctly, is a powerful tool in your mathematical arsenal, allowing you to conquer inequality challenges with confidence. Always remember to consider the sign of the number you're multiplying or dividing by, as this determines whether the inequality sign needs to be flipped. This attention to detail is what separates a good mathematician from a great one.

The Answer Unveiled

Drumroll, please! Based on our step-by-step analysis, the property used in the second step of solving the inequality is the Multiplication Property (B). We divided both sides of the inequality by 5 to isolate 'x,' and this is a direct application of the multiplication property. It's like using the right tool for the job – in this case, division to undo multiplication.

The multiplication property is crucial for solving inequalities because it allows us to isolate the variable we're trying to solve for. It's a fundamental concept in algebra, and mastering it will unlock your ability to solve a wide range of problems. Remember the key: if you multiply or divide by a negative number, flip the inequality sign. But when the number is positive, you can proceed without flipping. This property isn't just about getting the right answer; it's about understanding the logic behind mathematical operations and how they affect relationships between numbers. By grasping this concept, you're building a strong foundation for more advanced math topics. Think of it as a building block – solid understanding here leads to success with more complex equations and inequalities later on. So, embrace the multiplication property, practice its application, and watch your problem-solving skills soar.

Why Not the Other Options?

It's always a good idea to understand why the other options aren't correct. This helps solidify your understanding of the topic and avoid common pitfalls.

A. Addition Property: We used the addition property in the first step when we added 9 to both sides. So, while it's a valid property, it's not the one we used in the second step.
C. Transitive Property: The transitive property is more about relating three quantities. For example, if a < b and b < c, then a < c. This property wasn't directly used in solving our inequality.

Wrapping Up

So, there you have it! We've successfully identified the property used in the second step of solving the inequality: the Multiplication Property. We've also explored why the other options weren't the right fit. Remember, math isn't just about memorizing steps; it's about understanding the underlying principles. Keep practicing, keep exploring, and you'll become a math whiz in no time! And always remember, the journey of understanding math is like unwrapping a gift – each layer you peel back reveals something new and exciting. So, keep unwrapping and keep learning!

This detailed explanation not only provides the answer but also reinforces the understanding of inequality properties, making you a more confident problem-solver. By breaking down each step and explaining the reasoning behind it, we've transformed a simple math problem into a valuable learning experience. This approach is key to mastering math – it's not just about the answer, it's about the journey of discovery and the deep understanding you gain along the way. So, embrace the challenge, ask questions, and never stop exploring the wonderful world of mathematics!