Rational Numbers: Find X And Prove Its Nature

Hey guys! Let's dive into this fascinating problem that combines fractions, integers, and the concept of rational numbers. We've got a real head-scratcher here, but don't worry, we'll break it down together, step-by-step, until it's crystal clear. So, buckle up and let's get started!

Understanding the Problem

The core of this problem lies in understanding how different types of numbers interact, especially when we're dealing with addition and the concept of rational numbers. Let’s quickly recap what each term means to make sure we're all on the same page.

• Integers: These are whole numbers – no fractions or decimals allowed! They can be positive (like 1, 2, 3…), negative (like -1, -2, -3…), or zero.
• Nonzero Integers: This simply means integers that are not zero. So, we're talking about numbers like -5, -2, 1, 7, and so on.
• Rational Numbers: This is where it gets interesting. A rational number is any number that can be expressed as a fraction ${\frac{p}{q}}$, where both p and q are integers, and q is not zero. Think of it as a ratio of two integers. Examples include ${\frac{1}{2}}$, ${\frac{-3}{4}}$, 5 (which can be written as ${\frac{5}{1}}$), and even terminating or repeating decimals like 0.75 (which is ${\frac{3}{4}}$).

Now, let's restate the problem in simpler terms. We're given four nonzero integers: a, b, c, and d. We have a fraction ${\frac{a}{b}}$, and we're adding some unknown number x to it. The result of this addition is another fraction ${\frac{c}{d}}$. Our mission, should we choose to accept it, is to figure out which statement proves that x must be a rational number.

To really nail this, we need to think about what it means for x to be rational. Remember, a number is rational if we can write it as a fraction with integers on top and bottom. So, our goal is to manipulate the given equation to isolate x and see if we can express it in that form. This involves using algebraic principles like isolating variables and performing operations on fractions, but don't let that scare you. We'll take it nice and slow.

By carefully dissecting the problem and understanding the definitions of integers and rational numbers, we've set the stage for finding the solution. The next step is to use our algebraic skills to isolate x and see what we can discover about its nature. Let's keep going!

Isolating x: The Algebraic Maneuver

Alright, let’s roll up our sleeves and get our hands dirty with some algebra! This is where we take the information we have and start rearranging things to get what we want. In this case, what we want is x all by itself on one side of the equation. This process is called isolating the variable, and it’s a fundamental skill in algebra.

We start with the equation given in the problem:

$$\frac{a}{b} + x = \frac{c}{d}$$

Our goal is to get x alone. To do this, we need to get rid of the ${\frac{a}{b}}$ term on the left side. The way we do that is by subtracting ${\frac{a}{b}}$ from both sides of the equation. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. It’s like a see-saw – if you take weight off one side, you have to take the same weight off the other side to keep it level.

So, we subtract ${\frac{a}{b}}$ from both sides:

$$x = \frac{c}{d} - \frac{a}{b}$$

Now, x is isolated! We’ve successfully maneuvered the equation to get x by itself. But we're not quite done yet. We need to show that x is a rational number. To do that, we need to combine the two fractions on the right side into a single fraction. This involves finding a common denominator.

Finding a common denominator is like speaking the same language. If two fractions have different denominators, they're speaking different languages. To add or subtract them, we need to translate them into the same language – that's the common denominator. The easiest common denominator to find is simply the product of the two denominators. In this case, the common denominator for ${\frac{c}{d}}$ and ${\frac{a}{b}}$ is bd.

To get ${\frac{c}{d}}$ to have a denominator of bd, we multiply both the numerator and the denominator by b:

$$\frac{c}{d} = \frac{c \times b}{d \times b} = \frac{bc}{bd}$$

Similarly, to get ${\frac{a}{b}}$ to have a denominator of bd, we multiply both the numerator and the denominator by d:

$$\frac{a}{b} = \frac{a \times d}{b \times d} = \frac{ad}{bd}$$

Now that both fractions have the same denominator, we can subtract them:

$$x = \frac{bc}{bd} - \frac{ad}{bd} = \frac{bc - ad}{bd}$$

And there we have it! We’ve expressed x as a single fraction. The numerator is bc - ad, and the denominator is bd. Now, let's think about what this means in terms of rational numbers.

Proving Rationality: Why x is Rational

We've arrived at a crucial point in our problem-solving journey. We've successfully isolated x and expressed it as a single fraction: ${\frac{bc - ad}{bd}}$. But how does this prove that x is a rational number? Let's break it down.

Remember our definition of a rational number: It's any number that can be written as a fraction ${\frac{p}{q}}$, where p and q are integers, and q is not zero. So, to prove that x is rational, we need to show that both the numerator (bc - ad) and the denominator (bd) are integers, and that the denominator is not zero.

The problem statement gives us a vital clue: a, b, c, and d are all nonzero integers. This is the key piece of information that unlocks the solution. Let’s think about what happens when we perform operations on integers.

• Multiplication: When you multiply two integers, you always get another integer. For example, 3 * 5 = 15, -2 * 4 = -8, and so on. So, bc and ad are both integers because they are the products of integers.
• Subtraction: When you subtract one integer from another, you also get an integer. For example, 7 - 2 = 5, -3 - 1 = -4, and so on. Therefore, bc - ad is an integer because it's the difference between two integers.
• Denominator: We already know that b and d are nonzero integers. When we multiply them (bd), we get another nonzero integer. This is crucial because the denominator of a fraction cannot be zero.

So, let's recap: bc - ad is an integer, and bd is a nonzero integer. This means that ${\frac{bc - ad}{bd}}$ perfectly fits the definition of a rational number! We've shown that x can be expressed as a fraction with an integer numerator and a nonzero integer denominator. Therefore, x must be a rational number.

This is where the statements given in the problem come into play. We need to match our derived expression for x with one of the given options. Remember, the journey of solving a math problem isn’t just about getting the right answer; it’s about understanding why the answer is correct. In this case, we've not only found the expression for x, but we've also rigorously proven why that expression guarantees that x is a rational number. Let’s move on to comparing our result with the answer choices.

Matching the Expression: Finding the Right Statement

Okay, we've done the hard work of isolating x and proving that it's a rational number. Now comes the final step: matching our derived expression for x with one of the statements provided in the problem. This is like finding the missing piece of a puzzle – we have the shape, and now we need to see which piece fits perfectly.

We found that:

$$x = \frac{bc - ad}{bd}$$

Let's take a look at the answer choices (which were not provided in the initial problem, but we'll consider the likely options):

A. ${x = \frac{c-a}{d-b}}$ B. ${x = \frac{c+a}{d-b}}$ C. ${x = \frac{c}{d} - \frac{a}{b}}$ D. ${x = \frac{bc - ad}{bd}}$

By carefully comparing our derived expression with the answer choices, we can see that option D is an exact match!

Option D, ${x = \frac{bc - ad}{bd}}$, is precisely the expression we derived by isolating x and combining the fractions. This statement directly shows that x is a rational number because it expresses x as a fraction with an integer numerator (bc - ad) and a nonzero integer denominator (bd).

The other options might look similar at first glance, but they are not equivalent to our derived expression. For example, options A and B have denominators of d - b, which is different from our denominator of bd. Option C, while representing the initial step in isolating x, doesn't explicitly show x as a single fraction, which is crucial for proving rationality.

Therefore, the correct statement that proves x must be a rational number is option D. We've not only found the right answer, but we've also understood why it's the right answer. We've traced the logical steps, from understanding the definitions of integers and rational numbers to performing algebraic manipulations and proving the rationality of x. This is the essence of problem-solving in mathematics – it's not just about the destination, but the journey itself!

Final Thoughts: Key Takeaways

Woohoo! We've successfully navigated this problem, and hopefully, you've gained some valuable insights along the way. Let's quickly recap the key takeaways from our mathematical adventure:

• Understanding Definitions: The foundation of solving any math problem lies in understanding the definitions of the terms involved. In this case, knowing what integers and rational numbers are was crucial.
• Isolating Variables: Algebraic manipulation, like isolating variables, is a powerful tool for solving equations. It allows us to rearrange equations to reveal hidden relationships and express unknowns in terms of knowns.
• Common Denominators: When adding or subtracting fractions, finding a common denominator is essential. It allows us to combine fractions into a single, manageable expression.
• Proving Rationality: To prove that a number is rational, we need to show that it can be expressed as a fraction with an integer numerator and a nonzero integer denominator.
• Step-by-Step Approach: Complex problems can be tackled by breaking them down into smaller, more manageable steps. Each step builds upon the previous one, leading us closer to the solution.

By mastering these concepts and techniques, you'll be well-equipped to tackle similar problems in the future. Remember, mathematics is not just about memorizing formulas; it's about developing a logical and analytical way of thinking. So, keep practicing, keep exploring, and keep having fun with math! You've got this!