Equivalent Rational Expression A Step By Step Solution

This article delves into the problem of identifying rational expressions, focusing on the given expression ${\frac{4}{x-3}}$ and determining which of the provided options is equivalent. We will explore the fundamental concepts of rational expressions, including multiplication and division, to methodically analyze each option. This comprehensive guide aims to provide a clear understanding of how to manipulate rational expressions and arrive at the correct solution. Understanding rational expressions is crucial in algebra and calculus, as they form the basis for more complex mathematical concepts. This article will not only help you solve this specific problem but also equip you with the knowledge to tackle similar problems with confidence.

Understanding Rational Expressions

Rational expressions are essentially fractions where the numerator and denominator are polynomials. Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. For example, ${x^2 + 3x - 5}$ and ${4x - 2}$ are polynomials, making ${\frac{x^2 + 3x - 5}{4x - 2}}$ a rational expression. The key to working with rational expressions lies in understanding how to simplify, multiply, and divide them, much like regular fractions. Simplification involves canceling out common factors from the numerator and denominator, while multiplication and division follow specific rules that we will explore in detail.

When dealing with rational expressions, it's crucial to identify any restrictions on the variable. These restrictions occur when the denominator of the expression equals zero, as division by zero is undefined. For the given expression, ${\frac{4}{x-3}}$, the denominator is ${x-3}$. Setting this equal to zero, we get ${x-3 = 0}$, which gives us ${x = 3}$. Therefore, ${x}$ cannot be equal to 3, as it would make the expression undefined. This restriction is important to keep in mind when manipulating and simplifying rational expressions. Understanding these restrictions ensures that the solutions we obtain are valid within the domain of the expression. By carefully considering the domain, we can avoid errors and arrive at accurate results. The ability to identify restrictions is a fundamental skill in working with rational expressions and is essential for solving more complex problems in algebra and calculus.

Analyzing Option A: ${\frac{x+2}{x-3} \cdot \frac{4}{x+2}}$

Let's examine option A: ${\frac{x+2}{x-3} \cdot \frac{4}{x+2}}$. To determine if this expression is equivalent to ${\frac{4}{x-3}}$, we need to perform the multiplication. When multiplying rational expressions, we multiply the numerators together and the denominators together. In this case, we have:

${ \frac{x+2}{x-3} \cdot \frac{4}{x+2} = \frac{(x+2) \cdot 4}{(x-3) \cdot (x+2)} }$

Now, we simplify the expression by canceling out common factors. We can see that ${(x+2)}$ appears in both the numerator and the denominator. Canceling out this common factor, we get:

${ \frac{(x+2) \cdot 4}{(x-3) \cdot (x+2)} = \frac{4}{x-3} }$

After simplification, the expression becomes ${\frac{4}{x-3}}$, which is the original expression we were given. Therefore, option A is equivalent to the given expression. This demonstrates the principle of multiplying rational expressions and simplifying by canceling common factors. By carefully multiplying and simplifying, we can determine if two rational expressions are equivalent. This process is fundamental to solving algebraic equations and simplifying complex expressions. The ability to recognize and cancel common factors is a key skill in working with rational expressions.

Analyzing Option B: ${\frac{x-3}{x+2} \cdot \frac{x+2}{4}}$

Now, let's analyze option B: ${\frac{x-3}{x+2} \cdot \frac{x+2}{4}}$. To determine if this expression is equivalent to ${\frac{4}{x-3}}$, we will follow the same procedure as before: multiply the numerators and the denominators, and then simplify. Multiplying the numerators and denominators, we get:

${ \frac{x-3}{x+2} \cdot \frac{x+2}{4} = \frac{(x-3)(x+2)}{(x+2)(4)} }$

Next, we look for common factors to cancel out. We can see that ${(x+2)}$ is a common factor in both the numerator and the denominator. Canceling out the ${(x+2)}$ term simplifies the expression to:

${ \frac{(x-3)(x+2)}{(x+2)(4)} = \frac{x-3}{4} }$

The simplified expression is ${\frac{x-3}{4}}$. Comparing this to the original expression ${\frac{4}{x-3}}$, we can see that they are not the same. The numerator and denominator are switched, and there is no way to manipulate ${\frac{x-3}{4}}$ to obtain ${\frac{4}{x-3}}$. Therefore, option B is not equivalent to the given expression. This analysis highlights the importance of careful multiplication and simplification when working with rational expressions. Even though there was a common factor that could be canceled, the resulting expression was not equivalent to the original.

Analyzing Option C: ${\frac{x+2}{x-3} \div \frac{4}{x+2}}$

Let's consider option C: ${\frac{x+2}{x-3} \div \frac{4}{x+2}}$. To determine if this expression is equivalent to ${\frac{4}{x-3}}$, we need to remember the rule for dividing rational expressions: dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, we can rewrite the division as a multiplication:

${ \frac{x+2}{x-3} \div \frac{4}{x+2} = \frac{x+2}{x-3} \cdot \frac{x+2}{4} }$

Now, we multiply the numerators and denominators:

${ \frac{x+2}{x-3} \cdot \frac{x+2}{4} = \frac{(x+2)(x+2)}{(x-3)(4)} }$

This simplifies to:

${ \frac{(x+2)^2}{4(x-3)} }$

Expanding the numerator gives us:

${ \frac{x^2 + 4x + 4}{4(x-3)} }$

This expression is clearly not equivalent to ${\frac{4}{x-3}}$. The numerator is a quadratic expression, and the denominator contains a linear term, making it impossible to simplify to the original expression. Therefore, option C is not the correct answer. This analysis underscores the importance of correctly applying the rules of division for rational expressions and carefully simplifying the resulting expression.

Analyzing Option D: ${\frac{x-3}{x+2} \div \frac{4}{x+2}}$

Finally, let's analyze option D: ${\frac{x-3}{x+2} \div \frac{4}{x+2}}$. As with option C, we need to remember that dividing by a fraction is the same as multiplying by its reciprocal. So, we rewrite the expression as a multiplication:

${ \frac{x-3}{x+2} \div \frac{4}{x+2} = \frac{x-3}{x+2} \cdot \frac{x+2}{4} }$

Now, we multiply the numerators and the denominators:

${ \frac{x-3}{x+2} \cdot \frac{x+2}{4} = \frac{(x-3)(x+2)}{(x+2)(4)} }$

We can see that ${(x+2)}$ is a common factor in both the numerator and the denominator. Canceling out the ${(x+2)}$ term simplifies the expression to:

${ \frac{(x-3)(x+2)}{(x+2)(4)} = \frac{x-3}{4} }$

The simplified expression is ${\frac{x-3}{4}}$, which is not equivalent to the original expression ${\frac{4}{x-3}}$. The numerator and denominator are switched, and there's no valid way to transform ${\frac{x-3}{4}}$ into ${\frac{4}{x-3}}$. Thus, option D is incorrect. This analysis reinforces the need for careful attention to detail when dividing rational expressions and simplifying the results. Although there was a common factor to cancel, the final expression did not match the original.

Conclusion

After a thorough analysis of all the options, we have determined that option A, ${\frac{x+2}{x-3} \cdot \frac{4}{x+2}}$, is the only expression equivalent to the given expression ${\frac{4}{x-3}}$. This was achieved by multiplying the rational expressions and simplifying by canceling out common factors. The other options, B, C, and D, resulted in different expressions after simplification, demonstrating that they are not equivalent to the original.

This exercise highlights the importance of understanding the fundamental operations of rational expressions, including multiplication and division, and the crucial role of simplification in determining equivalence. By mastering these concepts, you can confidently tackle more complex algebraic problems involving rational expressions. Remember to always look for common factors to cancel and to apply the rules of multiplication and division correctly. This comprehensive approach ensures accuracy and a deeper understanding of the underlying mathematical principles.