Evaluate 4(x+3)(x+1) / (x+5)(x-5) For X=3 A Step-by-Step Solution

In the realm of mathematics, evaluating expressions is a fundamental skill. Algebraic expressions, combinations of variables, constants, and mathematical operations, are the building blocks of equations and formulas. The ability to evaluate these expressions accurately is crucial for solving mathematical problems and understanding various concepts. This article will guide you through the process of evaluating a specific algebraic expression for a given value of the variable, providing a clear and concise explanation of each step involved. We will focus on the expression $\frac{4(x+3)(x+1)}{(x+5)(x-5)}$ and evaluate it when $x = 3$. This example will not only demonstrate the evaluation process but also highlight the importance of order of operations and careful substitution. Mastering the evaluation of algebraic expressions is not just about finding the correct numerical answer; it's about developing a deeper understanding of how mathematical relationships work and building a solid foundation for more advanced mathematical concepts. As we proceed, we will emphasize clarity and precision, ensuring that each step is easy to follow and understand. So, let's embark on this mathematical journey and explore the intricacies of evaluating expressions.

The problem at hand involves evaluating the algebraic expression $\frac{4(x+3)(x+1)}{(x+5)(x-5)}$ for the specific value of $x = 3$. This task requires us to substitute the value of $x$ into the expression and then simplify it according to the order of operations. The expression consists of several factors, both in the numerator and the denominator, which means we need to handle the substitution and simplification carefully to avoid any errors. The numerator contains the product of 4, $(x+3)$, and $(x+1)$, while the denominator contains the product of $(x+5)$ and $(x-5)$. This structure necessitates a systematic approach, where we first substitute $x = 3$ into each factor and then perform the multiplications and divisions in the correct order. The goal is to arrive at a single numerical value that represents the expression's result when $x$ is 3. This type of problem is common in algebra and serves as a foundational exercise for more complex algebraic manipulations. Successfully solving this problem requires not only the knowledge of algebraic principles but also attention to detail and a methodical approach. By breaking down the expression and substituting the value step by step, we can ensure an accurate evaluation.

To begin the evaluation, we substitute $x = 3$ into the expression $\frac{4(x+3)(x+1)}{(x+5)(x-5)}$. This involves replacing every instance of $x$ in the expression with the number 3. Let's break down the substitution process step by step:

1. Numerator:
   • Substitute $x = 3$ into $(x + 3)$: $(3 + 3) = 6$
   • Substitute $x = 3$ into $(x + 1)$: $(3 + 1) = 4$
   • The numerator becomes $4 * 6 * 4$
2. Denominator:
   • Substitute $x = 3$ into $(x + 5)$: $(3 + 5) = 8$
   • Substitute $x = 3$ into $(x - 5)$: $(3 - 5) = -2$
   • The denominator becomes $8 * (-2)$

After the substitution, the expression now looks like this: $\frac{4 * 6 * 4}{8 * (-2)}$. This simplified form allows us to perform the arithmetic operations to arrive at the final answer. The key to this step is to be meticulous in replacing each $x$ with its value and keeping track of the operations. This careful substitution is crucial for avoiding errors and ensuring an accurate evaluation of the expression. In the next step, we will simplify this numerical expression by performing the multiplications and division.

Now that we have substituted $x = 3$ into the expression, we have $\frac{4 * 6 * 4}{8 * (-2)}$. The next step is to simplify this numerical expression by performing the arithmetic operations. We will follow the order of operations, which dictates that we perform multiplications before division.

1. Numerator:
   • Multiply the numbers in the numerator: $4 * 6 * 4 = 96$
2. Denominator:
   • Multiply the numbers in the denominator: $8 * (-2) = -16$

After performing the multiplications, the expression simplifies to $\frac{96}{-16}$. Now, we perform the division to get the final result. Divide 96 by -16:

• $\frac{96}{-16} = -6$

Therefore, the simplified value of the expression is -6. This step demonstrates the importance of following the order of operations to arrive at the correct answer. By first multiplying the numbers in the numerator and the denominator separately and then performing the division, we have successfully simplified the expression. The result, -6, represents the value of the original algebraic expression when $x = 3$. This process of simplification is a critical skill in algebra and is used extensively in solving equations and simplifying more complex expressions.

After performing the substitution and simplification, we have arrived at the final result. The value of the expression $\frac{4(x+3)(x+1)}{(x+5)(x-5)}$ when $x = 3$ is -6. This means that when we replace $x$ with 3 in the given expression and perform the necessary arithmetic operations, the outcome is -6. This result is a single numerical value that represents the expression's worth for the specified value of the variable. The process of arriving at this result involved several steps, including substituting the value of $x$, multiplying the factors in the numerator and denominator, and finally, dividing the numerator by the denominator. Each of these steps was crucial in ensuring the accuracy of the final answer. The final result, -6, is not just a number; it's the solution to the problem, and it represents a specific point on the number line. Understanding how to evaluate expressions like this is fundamental to algebra and is essential for solving more complex mathematical problems. In the next section, we will briefly recap the steps taken and highlight the key concepts involved in this evaluation process.

In conclusion, evaluating the expression $\frac{4(x+3)(x+1)}{(x+5)(x-5)}$ for $x = 3$ yielded a final result of -6. This process involved several key steps, each of which is crucial for accurate evaluation. First, we substituted $x = 3$ into the expression, replacing every instance of $x$ with the number 3. This step transformed the algebraic expression into a numerical expression. Next, we simplified the expression by performing the arithmetic operations according to the order of operations. This involved multiplying the factors in the numerator and the denominator separately and then dividing the numerator by the denominator. The order of operations is a critical concept in mathematics, ensuring that we perform operations in the correct sequence to arrive at the correct answer. The final step was to arrive at the simplified value, which in this case was -6. This result represents the value of the expression when $x$ is equal to 3. Evaluating algebraic expressions is a fundamental skill in mathematics, and mastering this skill is essential for solving more complex problems. This exercise has demonstrated the importance of careful substitution, adherence to the order of operations, and meticulous simplification. By following these steps, we can confidently evaluate a wide range of algebraic expressions. Understanding these concepts not only helps in solving mathematical problems but also in building a strong foundation for more advanced mathematical studies.

The correct answer is B. -6.