Simplify Complex Fractions: A Step-by-Step Guide
Hey guys! Let's dive into this intriguing mathematical expression and break it down step by step. We've got a complex fraction here, and trust me, it looks more intimidating than it actually is. Our goal is to simplify this beast and understand the relationships between the variables. So, grab your metaphorical mathematical shovels, and let's get digging!
Understanding the Beast: The Initial Expression
Our journey begins with the complex fraction: (1.2x + 11) / [2(2z + y) - 5] ÷ [3|(y + z) - x| / 4]. At first glance, it might seem like a jumbled mess of variables and numbers. But don't worry, we'll dissect it piece by piece. The key here is to remember the order of operations (PEMDAS/BODMAS) and how fractions within fractions work. We're dealing with a fraction divided by another fraction, and that's where the fun begins. Let’s first focus on the numerator and denominator separately to make things clearer.
The numerator of the main fraction is (1.2x + 11) / [2(2z + y) - 5]. This part has its own internal fraction. The numerator of this inner fraction is a simple linear expression: 1.2x + 11. It represents a straight line if you were to graph it, with a slope of 1.2 and a y-intercept of 11. The denominator of this inner fraction is a bit more interesting: 2(2z + y) - 5. It involves variables z and y, and we'll need to simplify it further. We'll distribute the 2 and then combine any like terms.
Now, let's shift our attention to the denominator of the main fraction: 3|(y + z) - x| / 4. This is another fraction, but this time it involves an absolute value. Remember, the absolute value of a number is its distance from zero, so it's always non-negative. The expression inside the absolute value, (y + z) - x, is what we need to evaluate first. Then, we take the absolute value of the result, multiply it by 3, and finally divide by 4. This part adds a layer of complexity because the absolute value can change the sign of the expression depending on the values of x, y, and z.
Before we proceed further, it’s crucial to acknowledge the domain restrictions imposed by this expression. The denominator of any fraction cannot be zero. Therefore, we need to ensure that both [2(2z + y) - 5] and [3|(y + z) - x| / 4] are not equal to zero. This gives us two conditions to consider: 2(2z + y) - 5 ≠0 and 3|(y + z) - x| / 4 ≠0. These conditions will help us identify any values of x, y, and z that would make the expression undefined.
Simplifying the Numerator: Taming the Top Half
Let's simplify the numerator of the main fraction: (1.2x + 11) / [2(2z + y) - 5]. The first step is to tackle the denominator of this inner fraction. We need to distribute the 2 inside the parentheses: 2(2z + y) - 5 becomes 4z + 2y - 5. Now, we have (1.2x + 11) / (4z + 2y - 5). There are no more like terms to combine in the denominator, so we've simplified it as much as possible for now. This simplified form allows us to see the relationship between the variables more clearly. The numerator remains 1.2x + 11, which is a simple linear expression.
The simplified numerator is (1.2x + 11) / (4z + 2y - 5). This form highlights the interplay between x, y, and z. The value of the entire numerator depends on the values of all three variables. It's crucial to remember that 4z + 2y - 5 cannot be equal to zero, as this would make the fraction undefined. This condition is a critical constraint on the possible values of y and z.
To further illustrate, let's consider a few examples. If x = 0, the numerator becomes 11 / (4z + 2y - 5). If z and y are such that 4z + 2y - 5 = 1, then the numerator equals 11. Conversely, if 4z + 2y - 5 = 11, the numerator becomes 1. These simple examples demonstrate how the values of the variables influence the overall value of the numerator. Understanding these relationships is essential for grasping the behavior of the entire complex fraction.
Simplifying the Denominator: Unveiling the Bottom Half
Now, let's simplify the denominator of the main fraction: 3|(y + z) - x| / 4. This part involves an absolute value, so we need to be a bit careful. The first step is to focus on the expression inside the absolute value: (y + z) - x. This expression represents the difference between the sum of y and z, and x. The absolute value of this difference, |(y + z) - x|, ensures that the result is always non-negative.
Next, we multiply the absolute value by 3: 3|(y + z) - x|. This scales the non-negative value by a factor of 3. Finally, we divide the result by 4: [3|(y + z) - x|] / 4. This gives us the simplified denominator. The absolute value term makes this expression interesting because it introduces cases where the sign of (y + z) - x matters. If (y + z) - x is positive, the absolute value doesn't change anything. But if (y + z) - x is negative, the absolute value makes it positive.
The simplified denominator is [3|(y + z) - x|] / 4. This form clearly shows the impact of the absolute value. For instance, if (y + z) - x = 5, then the denominator is (3 * 5) / 4 = 15/4. If (y + z) - x = -5, the absolute value changes it to 5, and the denominator is still 15/4. This symmetry is a key characteristic of absolute value expressions. The denominator is always non-negative, which is a crucial piece of information when we consider the overall fraction.
Consider a scenario where x = 1, y = 2, and z = -1. Then (y + z) - x = (2 + (-1)) - 1 = 0. The absolute value of 0 is 0, so the denominator becomes (3 * 0) / 4 = 0. This is a critical case because it makes the entire main fraction undefined. This example highlights the importance of understanding when the denominator can be zero and avoiding those values.
Dividing Fractions: The Grand Finale
We've simplified the numerator to (1.2x + 11) / (4z + 2y - 5) and the denominator to [3|(y + z) - x|] / 4. Now, we need to divide the numerator by the denominator. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, we flip the denominator and multiply.
This means we'll multiply (1.2x + 11) / (4z + 2y - 5) by 4 / [3|(y + z) - x|]. The resulting expression is: [(1.2x + 11) * 4] / [(4z + 2y - 5) * 3|(y + z) - x|]. This is the simplified form of the original complex fraction. We've combined the numerators and the denominators, and now we have a single fraction.
The final simplified expression is [4(1.2x + 11)] / [3(4z + 2y - 5)|(y + z) - x|]. This form reveals the overall structure of the fraction and how the variables interact. The numerator is a linear expression in x, scaled by a factor of 4. The denominator involves a product of two terms: (4z + 2y - 5) and the absolute value term |(y + z) - x|. This denominator is more complex and has a significant impact on the behavior of the fraction.
To make this even clearer, let's distribute the constants in the numerator: 4(1.2x + 11) = 4.8x + 44. So, the final expression can also be written as (4.8x + 44) / [3(4z + 2y - 5)|(y + z) - x|]. This form is equivalent to the previous one but might be easier to work with in some cases. Remember, the key to simplifying complex expressions is to break them down into smaller, manageable parts and then combine them carefully.
Conclusion: Mastering the Complex Fraction
Wow, guys, we've made it! We successfully navigated through the complex fraction (1.2x + 11) / [2(2z + y) - 5] ÷ [3|(y + z) - x| / 4] and simplified it to [4(1.2x + 11)] / [3(4z + 2y - 5)|(y + z) - x|]. This journey involved understanding the order of operations, dealing with fractions within fractions, and handling absolute values. We also identified the critical conditions where the denominator cannot be zero.
This exercise highlights the importance of breaking down complex problems into smaller, more manageable steps. By simplifying the numerator and denominator separately, and then combining them using the rules of fraction division, we were able to arrive at a much clearer expression. The final simplified form allows us to see the relationships between the variables x, y, and z more easily. It also underscores the significance of considering domain restrictions to avoid undefined expressions.
So, the next time you encounter a complex fraction, remember the techniques we've discussed here. Break it down, simplify each part, and carefully combine the results. With practice and a solid understanding of the fundamentals, you'll be able to tackle even the most intimidating mathematical expressions. Keep exploring, keep questioning, and keep simplifying! You've got this!