Correct Representations Of Inequality -3(2x-5) < 5(2-x)

Hey guys! Today, we're diving into the fascinating world of inequalities, specifically tackling the problem: Which are the correct representations of the inequality -3(2x-5) < 5(2-x)? Select two options.

This isn't just about crunching numbers; it's about understanding the fundamental principles that govern these mathematical relationships. We'll break down the inequality step-by-step, exploring the various transformations we can apply while preserving its integrity. This will not only help you solve this particular problem but also equip you with the skills to confidently tackle any inequality that comes your way.

Decoding the Inequality -3(2x-5) < 5(2-x)

When faced with an inequality like -3(2x-5) < 5(2-x), the first step is to simplify both sides. This involves applying the distributive property, which, if you remember, is the rule that lets us multiply a single term by multiple terms inside parentheses. Let's take a closer look at how this works in our specific case. Remember, our main keyword here is inequality representations, so we'll be sure to highlight how each step transforms the original inequality.

Applying the Distributive Property: A Crucial First Step

On the left side of the inequality, we have -3(2x-5). To distribute the -3, we multiply it by each term inside the parentheses. So, -3 multiplied by 2x gives us -6x, and -3 multiplied by -5 gives us +15 (remember, a negative times a negative is a positive!). Therefore, the left side becomes -6x + 15. It's super important to get the signs right here, as a small mistake can change the whole outcome of the problem. Think of it like a recipe – if you add too much of one ingredient, the whole dish can be ruined! The distributive property is a cornerstone of algebraic manipulation, and mastering it is crucial for solving not just inequalities, but also equations and many other mathematical problems. When dealing with inequality representations, understanding how operations like distribution affect the inequality is key.

Similarly, on the right side of the inequality, we have 5(2-x). Distributing the 5, we multiply it by 2, which gives us 10, and by -x, which gives us -5x. So, the right side becomes 10 - 5x. Again, pay close attention to the signs! It's easy to make a mistake if you rush through this step, and accuracy is paramount in math. By applying the distributive property correctly, we've taken a big step toward simplifying our inequality and making it easier to solve. Now, we have a new, but equivalent, inequality representation to work with.

The Transformed Inequality: -6x + 15 < 10 - 5x

After applying the distributive property to both sides, our original inequality -3(2x-5) < 5(2-x) has been transformed into -6x + 15 < 10 - 5x. This is a crucial step in finding the correct inequality representations. Notice how this new form looks different from the original, but it's actually mathematically equivalent. This means that both inequalities have the same solution set – they represent the same relationship between x and the other numbers. This is a fundamental concept in algebra: we can manipulate equations and inequalities using valid operations without changing their underlying meaning. Think of it like translating a sentence from one language to another; the words change, but the meaning stays the same. The key here is that the inequality representation has changed, but the solution set remains constant. This is a core principle when simplifying and solving inequalities.

This simplified form, -6x + 15 < 10 - 5x, is one of the options presented in the problem, making it one of the correct representations. We've already identified one of the two correct answers! But our journey doesn't end here. We need to find the other correct representation, which will likely involve further simplification and manipulation of the inequality. This process of transforming the inequality while preserving its solution set is the heart of solving these types of problems. It’s all about finding different inequality representations that ultimately lead to the same answer.

Isolating the Variable: Getting 'x' by Itself

Now that we've simplified the inequality to -6x + 15 < 10 - 5x, the next goal is to isolate the variable x on one side of the inequality. This is a common strategy in solving equations and inequalities, as it allows us to determine the range of values that satisfy the relationship. To do this effectively, we'll use the properties of inequalities, which are similar to the properties of equality but with a few important differences. We want to find the inequality representations where x is isolated.

Strategic Rearrangement: Moving Terms Across the Inequality

To isolate x, we need to gather all the x terms on one side of the inequality and all the constant terms (the numbers without x) on the other side. A common approach is to move the x terms to the side that will result in a positive coefficient for x. This can help avoid potential errors when dealing with negative signs. In our case, we have -6x + 15 < 10 - 5x. We can add 5x to both sides of the inequality. Remember, adding the same value to both sides of an inequality preserves the inequality. Think of it like a seesaw – if you add the same weight to both sides, it remains balanced. This gives us -6x + 5x + 15 < 10 - 5x + 5x, which simplifies to -x + 15 < 10. The act of moving terms around while maintaining the inequality is crucial for finding equivalent inequality representations.

Next, we need to move the constant term, 15, to the right side of the inequality. We can do this by subtracting 15 from both sides. Again, subtracting the same value from both sides preserves the inequality. This gives us -x + 15 - 15 < 10 - 15, which simplifies to -x < -5. We are getting closer to a solution, another inequality representation, where x is isolated! This step highlights the importance of performing the same operation on both sides to maintain the balance of the inequality. These steps are fundamental for manipulating and understanding inequality representations.

Dealing with the Negative Coefficient: A Critical Step

We're almost there, but we have a slight problem: our x term has a negative coefficient (-1). We have -x < -5. To get a positive x, we need to multiply (or divide) both sides of the inequality by -1. Now, here's a crucial rule to remember when dealing with inequalities: 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 a key difference between solving inequalities and solving equations. Think of it like looking in a mirror – everything is flipped! This sign change is absolutely critical for maintaining the correct relationship and finding accurate inequality representations.

So, multiplying both sides of -x < -5 by -1, and flipping the inequality sign, we get x > 5. This is a common mistake point, so always double-check this step! This new inequality representation, x > 5, tells us that the solution set includes all values of x that are greater than 5. However, this isn't one of the options provided in the question. This means we need to revisit our steps and see if we can derive one of the given options from our current form, or from an earlier inequality representation we found.

Connecting the Dots: Finding the Second Correct Representation

We've successfully isolated x and found that x > 5. However, this isn't one of the options given. Don't worry! This doesn't mean we've done anything wrong. It just means we need to look back at our previous steps to see if any of the intermediate forms of the inequality match the remaining options. Often, the correct answer is not the fully simplified form, but an equivalent form from an earlier stage. Remember, we are looking for inequality representations, and several forms might be correct.

Revisiting the Simplified Form: A Potential Key

Let's go back to the simplified form we obtained after applying the distributive property: -6x + 15 < 10 - 5x. We already identified this as one of the correct inequality representations. Now, let's compare this to the remaining options. If we look closely, we'll notice that this form exactly matches one of the options provided in the question! This reinforces the importance of carefully checking each step and recognizing equivalent forms of the inequality.

The Final Answer: Identifying the Correct Duo

We've successfully identified the two correct representations of the inequality -3(2x-5) < 5(2-x). The first is the simplified form obtained after applying the distributive property: -6x + 15 < 10 - 5x. The second, after isolating the variable and handling the negative coefficient, leads us to the solution x > 5. By carefully applying the rules of algebra and paying attention to the nuances of inequalities, we've navigated this problem and found the correct inequality representations.

Key Takeaways: Mastering Inequality Representations

This problem highlights several key concepts in working with inequalities. Firstly, the distributive property is essential for simplifying expressions and transforming inequalities into more manageable forms. Secondly, remember the golden rule: When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality sign. Finally, always be prepared to revisit previous steps and compare intermediate forms with the given options. Understanding inequality representations is about recognizing equivalent forms and manipulating them to reach a solution.

By mastering these techniques, you'll be well-equipped to tackle a wide range of inequality problems. Keep practicing, and remember that every problem is an opportunity to strengthen your understanding of these fundamental mathematical concepts. You got this!