Simplifying Rational Expressions Subtracting (r + 15)/7 - (r - 15)/7

Hey guys! Today, we are diving into the world of rational expressions, specifically focusing on subtraction. This might sound intimidating, but don't worry, we'll break it down step by step. We're going to tackle an expression that looks like this: (r + 15)/7 - (r - 15)/7. Our mission is to simplify this expression and express the final answer in its lowest terms. So, grab your pencils, and let's get started on this mathematical adventure! Understanding rational expressions is crucial as it forms the foundation for more advanced algebraic concepts. When you master this, you will be able to solve complex equations and simplify mathematical models in various fields, from physics to finance. This guide is designed to not only give you the answer, but also to equip you with the knowledge and confidence to tackle similar problems in the future. Remember, math isn't just about getting the correct answer; it's about understanding the process and logic behind it. Let’s embark on this journey together and make math a little less mysterious and a lot more fun!

Understanding Rational Expressions

Before we jump into the subtraction process, let's take a moment to understand what rational expressions are. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Think of it as a fancy way of saying a fraction with algebraic expressions on top and bottom.

In our case, we have two rational expressions: (r + 15)/7 and (r - 15)/7. Notice that both expressions have the same denominator, which is 7. This is a crucial observation because it makes the subtraction process much simpler. When subtracting fractions, having a common denominator is key. It's like making sure everyone's playing on the same field before starting the game. If the denominators were different, we’d first need to find a common denominator, but luckily for us, we can skip that step this time. Rational expressions are ubiquitous in mathematics, appearing in various contexts such as calculus, algebra, and even real-world applications like modeling rates and proportions. They allow us to represent relationships between quantities in a concise and powerful way. By understanding their structure and how to manipulate them, we can solve complex problems and gain deeper insights into mathematical concepts. So, let's keep this understanding in mind as we move forward and tackle the subtraction of these expressions.

Step-by-Step Subtraction Process

Now, let’s get to the heart of the matter: subtracting our rational expressions. Since we already have a common denominator, the process becomes quite straightforward. Here’s how we do it:

1. Combine the Numerators: When subtracting fractions with a common denominator, we simply subtract the numerators while keeping the denominator the same. So, we rewrite the expression as:

(r + 15) - (r - 15) / 7

It’s super important to pay attention to the signs here. The minus sign in front of the second fraction applies to the entire numerator (r - 15).

2. Distribute the Negative Sign: This is where things can get a little tricky if we're not careful. We need to distribute the negative sign to both terms inside the parentheses in the second numerator. This means we change the sign of 'r' and '-15'. Our expression now looks like this:

r + 15 - r + 15 / 7

Notice how the '-r' term comes from distributing the negative sign to the 'r' inside the parentheses, and the '+15' comes from distributing the negative sign to the '-15' (a negative times a negative is a positive!).

3. Combine Like Terms: Next, we need to simplify the numerator by combining like terms. In this case, we have 'r' and '-r', which cancel each other out, and we have '+15' and '+15', which combine to give us 30. So, the expression becomes:

30 / 7

4. Check for Simplification: Finally, we need to check if our fraction can be simplified further. In this case, 30 and 7 do not share any common factors other than 1, so the fraction is already in its simplest form. Therefore, our final answer is 30/7. Each step in this process is vital and builds upon the previous one. By understanding the logic behind each step, you're not just memorizing a procedure, but actually learning how to manipulate mathematical expressions. This skill is invaluable as you progress in your mathematical studies.

Detailed Explanation of Each Step

To really nail this down, let's dive deeper into each step of the subtraction process.

Combining the Numerators

The first step, combining the numerators, is all about setting up the problem correctly. We recognize that both fractions have the same denominator, which allows us to perform the subtraction directly on the numerators. This is similar to subtracting regular fractions where you have a common denominator, like 3/5 - 1/5. You simply subtract the numerators (3 - 1) and keep the denominator the same, resulting in 2/5. With rational expressions, the principle is the same, but we are dealing with algebraic expressions instead of simple numbers. The key takeaway here is that you can only combine numerators when the denominators are the same. If they aren't, you would need to find a common denominator first, which we’ll cover in another discussion. But for this problem, we’re in luck because we can proceed directly to subtracting the numerators. This step lays the foundation for the rest of the solution, so it's crucial to get it right.

Distributing the Negative Sign

Distributing the negative sign is a crucial step where many students often make mistakes, so let’s pay close attention. The negative sign in front of the parentheses acts like a multiplier for every term inside the parentheses. It's like saying we're subtracting the entire expression (r - 15), not just the 'r' part. Think of it as multiplying the entire expression inside the parentheses by -1. When we multiply -1 by 'r', we get '-r'. And when we multiply -1 by '-15', we get '+15' because a negative times a negative is a positive. This distribution is based on the distributive property of multiplication over addition and subtraction, which is a fundamental concept in algebra. If we skip this step or apply it incorrectly, we'll end up with the wrong answer. So, always double-check that you’ve distributed the negative sign to each term inside the parentheses. This seemingly small step can make a huge difference in the final result.

Combining Like Terms

After distributing the negative sign, the next step is to combine like terms. This is where we simplify the expression by grouping terms that have the same variable and exponent. In our case, we have 'r' and '-r', which are like terms because they both have the variable 'r' raised to the power of 1. When we combine them, r - r equals 0, effectively canceling each other out. Then, we have '+15' and '+15', which are also like terms because they are constants. When we combine them, 15 + 15 equals 30. This process of combining like terms is based on the commutative and associative properties of addition, which allow us to rearrange and group terms in any order. By combining like terms, we reduce the complexity of the expression, making it easier to understand and work with. This step is essential for simplifying algebraic expressions and solving equations.

Checking for Simplification

Our final step is to check if the fraction can be simplified further. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In other words, we can't divide both the numerator and the denominator by the same number to get smaller whole numbers. For example, the fraction 4/6 can be simplified to 2/3 because both 4 and 6 are divisible by 2. However, in our case, we have 30/7. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, while the factors of 7 are 1 and 7. The only common factor they share is 1, which means the fraction is already in its simplest form. This final check ensures that our answer is expressed in the most concise and clear way possible. It's like giving your final answer a polish to make sure it shines. Always remember to check for simplification to complete the problem correctly.

Common Mistakes to Avoid

To ensure you ace these types of problems, let's quickly go over some common mistakes people make and how to avoid them:

• Forgetting to Distribute the Negative Sign: This is the most common pitfall. Remember, the negative sign applies to the entire numerator, not just the first term. Always double-check that you've distributed it correctly.

• Incorrectly Combining Like Terms: Make sure you're only combining terms that are actually alike. For example, you can't combine 'r' with a constant number. They're not the same!

• Not Simplifying the Final Answer: Always check if your fraction can be simplified further. Leaving your answer in a non-simplified form isn't the end of the world, but it's like leaving a job half-finished.

• Rushing Through the Steps: Math requires precision. Rushing can lead to silly mistakes. Take your time, double-check your work, and make sure each step is accurate. By being aware of these common mistakes, you can actively work to avoid them. Math is a skill that improves with practice, so the more problems you solve, the better you'll become at spotting and avoiding these errors.

Real-World Applications

You might be wondering,