Simplifying The Difference Between (x+5)/(x+2) And (x+1)/(x^2+2x)

In the realm of mathematics, particularly algebra, dealing with rational expressions is a common task. These expressions, which are essentially fractions with polynomials in the numerator and denominator, often require simplification, addition, subtraction, multiplication, or division. This article delves into the process of finding the difference between two such rational expressions: ${\frac{x+5}{x+2}}$ and ${\frac{x+1}{x^2+2x}}$. We will break down each step, ensuring a clear understanding of the underlying principles and techniques involved. This comprehensive guide aims to equip you with the knowledge and skills necessary to tackle similar problems with confidence. The focus will be on manipulating algebraic fractions, identifying common denominators, and simplifying the final result to its most basic form. By the end of this exploration, you will not only understand how to solve this specific problem but also grasp the general methodology applicable to a wide range of algebraic fraction manipulations. This will significantly enhance your problem-solving capabilities in algebra and related mathematical fields.

1. Rewriting the problem

The initial step in solving the problem ${\frac{x+5}{x+2}-\frac{x+1}{x^2+2x}}$ involves rewriting the second fraction to identify common factors. The denominator ${x^2 + 2x}$ can be factored, which is a crucial step in finding a common denominator for the two fractions. Factoring the denominator allows us to see the underlying structure and identify common terms that can be used to simplify the expression. We begin by observing that both terms in the denominator, ${x^2}$ and ${2x}$, have a common factor of ${x}$. By factoring out this ${x}$, we transform the denominator into a product of simpler terms, which significantly aids in the subsequent steps of the problem-solving process. This is a standard technique in algebra, where complex expressions are broken down into simpler, more manageable components. This transformation not only simplifies the current problem but also builds a foundation for more advanced algebraic manipulations. The ability to recognize and factor common terms is a fundamental skill in algebra, enabling the simplification of expressions and the solution of equations. This step is not just about rewriting the expression; it is about unveiling its hidden structure and setting the stage for further simplification. By carefully factoring the denominator, we pave the way for finding a common denominator and ultimately subtracting the two fractions.

To do this, we factor ${x}$ from ${x^2 + 2x}$, resulting in ${x(x+2)}$. The problem can now be expressed as:

${ \frac{x+5}{x+2} - \frac{x+1}{x(x+2)} }$

This transformation is a key step because it reveals the common factor of ${(x+2)}$ in the denominators of both fractions. Identifying this common factor is essential for finding a common denominator, which is a prerequisite for subtracting the fractions. Without this factorization, it would be more challenging to determine the necessary steps to combine the two fractions. The ability to factor polynomials is a fundamental skill in algebra, and this step demonstrates its importance in simplifying complex expressions. By rewriting the expression in this factored form, we have made the next steps in the solution process much clearer and more manageable. This approach highlights the power of algebraic manipulation in transforming complex problems into simpler, more solvable forms. Factoring is not just a mechanical process; it is a strategic tool that unlocks the underlying structure of algebraic expressions.

2. Finding the Common Denominator

Now that we have rewritten the problem as ${\frac{x+5}{x+2} - \frac{x+1}{x(x+2)}}$, the next critical step is to find a common denominator. Finding a common denominator is essential when adding or subtracting fractions, as it allows us to combine the numerators over a single denominator. In this case, the denominators are ${(x+2)}$ and ${x(x+2)}$. To find the least common denominator (LCD), we need to identify the factors present in both denominators and include each factor the greatest number of times it appears in any denominator. The first denominator, ${(x+2)}$, has one factor: ${(x+2)}$. The second denominator, ${x(x+2)}$, has two factors: ${x}$ and ${(x+2)}$. Comparing these, we see that the LCD must include both ${x}$ and ${(x+2)}$. Therefore, the least common denominator is ${x(x+2)}$. This is because it includes all the factors from both original denominators, ensuring that both fractions can be expressed with this common denominator. This step is crucial for the subsequent subtraction, as it sets the stage for combining the numerators. Understanding how to find the LCD is a fundamental skill in working with fractions, both numerical and algebraic. It allows us to perform arithmetic operations on fractions in a consistent and accurate manner. This process is not just about finding a common denominator; it is about creating a unified framework within which we can manipulate and simplify fractional expressions.

To achieve this common denominator, we need to multiply the first fraction, ${\frac{x+5}{x+2}}$, by ${\frac{x}{x}}$. This gives us:

${ \frac{x+5}{x+2} \cdot \frac{x}{x} = \frac{x(x+5)}{x(x+2)} }$

Multiplying by ${\frac{x}{x}}$ is equivalent to multiplying by 1, which does not change the value of the fraction but allows us to rewrite it with the desired denominator. This is a fundamental technique in fraction manipulation. By multiplying both the numerator and the denominator by the same factor, we maintain the fraction's value while altering its form. This is essential for creating equivalent fractions that share a common denominator. In this specific case, multiplying by ${\frac{x}{x}}$ introduces the necessary factor of ${x}$ in the denominator, making it identical to the denominator of the second fraction. This step is a critical bridge that allows us to perform the subtraction. It highlights the flexibility and power of algebraic manipulation in transforming expressions to suit our needs. The ability to recognize the missing factors and introduce them through multiplication is a key skill in simplifying and solving algebraic problems. This process is not just about finding a common denominator; it is about strategically modifying expressions to facilitate further operations.

3. Performing the Subtraction

With the fractions now having a common denominator, we can proceed with the subtraction. The expression is now:

${ \frac{x(x+5)}{x(x+2)} - \frac{x+1}{x(x+2)} }$

To subtract fractions with a common denominator, we subtract the numerators and keep the denominator the same. Therefore, we have:

${ \frac{x(x+5) - (x+1)}{x(x+2)} }$

This step is a direct application of the rules of fraction arithmetic. Once a common denominator is established, the subtraction of fractions becomes a straightforward process of combining the numerators. It is crucial to maintain the common denominator throughout this step, as it provides the foundation for the combined fraction. The subtraction of the numerators must be performed carefully, paying attention to the signs and the distribution of terms. This is a common area for errors, so meticulous attention to detail is essential. The ability to combine fractions in this way is a fundamental skill in algebra and is essential for solving a wide range of problems. This step is not just about subtracting fractions; it is about synthesizing the individual components into a unified expression. By combining the numerators over the common denominator, we create a single fraction that represents the difference between the original two fractions.

Next, we expand the numerator:

${ \frac{x^2 + 5x - x - 1}{x(x+2)} }$

Expanding the numerator involves applying the distributive property and multiplying out the terms. This step is essential for simplifying the expression and combining like terms. The expansion of ${x(x+5)}$ to ${x^2 + 5x}$ is a straightforward application of the distributive property. However, the subtraction of ${(x+1)}$ requires careful attention to the negative sign. It is crucial to distribute the negative sign to both terms inside the parentheses, resulting in ${-x - 1}$. This is a common area for errors, so meticulous attention to detail is essential. The ability to expand and simplify algebraic expressions is a fundamental skill in algebra and is essential for solving equations and simplifying complex expressions. This step is not just about expanding the numerator; it is about preparing the expression for further simplification by eliminating parentheses and combining like terms. By carefully expanding the numerator, we set the stage for the next step, which involves combining like terms and simplifying the expression.

4. Simplifying the Expression

After expanding the numerator, we simplify the expression by combining like terms:

${ \frac{x^2 + 4x - 1}{x(x+2)} }$

In this step, we combine the ${5x}$ and ${-x}$ terms in the numerator, resulting in ${4x}$. This is a straightforward application of the rules of algebraic simplification. Combining like terms is a fundamental skill in algebra and is essential for reducing expressions to their simplest form. This process involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. In this case, the ${5x}$ and ${-x}$ terms are like terms, and their coefficients can be combined. This simplification step makes the expression more concise and easier to work with. It also allows us to see the overall structure of the expression more clearly. The ability to simplify algebraic expressions is a key skill in algebra and is essential for solving equations and simplifying complex expressions. This step is not just about combining like terms; it is about streamlining the expression and making it more manageable for further analysis or manipulation.

The simplified form of the expression is:

${ \frac{x^2 + 4x - 1}{x(x+2)} }$

This is the final simplified form of the expression, as the numerator cannot be factored further, and there are no common factors between the numerator and the denominator that can be canceled. The result represents the difference between the two original fractions in its most concise form. It is important to note that simplification is a crucial step in solving algebraic problems. It allows us to express the solution in a clear and understandable way. In this case, the simplified expression provides a direct representation of the difference between the two original fractions. This final form is not only the solution to the problem but also a testament to the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. The ability to simplify expressions to their final form is a key skill in algebra and is essential for solving a wide range of problems. This step is not just about finding the answer; it is about presenting the solution in its most elegant and understandable form.

5. Final Answer

Therefore, the difference between ${\frac{x+5}{x+2}}$ and ${\frac{x+1}{x^2+2x}}$ is:

${ \frac{x^2 + 4x - 1}{x(x+2)} }$

This is the final answer, representing the simplified form of the original expression. It is the culmination of all the steps we have taken, from rewriting the problem to finding a common denominator, performing the subtraction, and simplifying the result. This answer provides a concise and accurate representation of the difference between the two original fractions. It is important to recognize that the final answer is not just a number or an expression; it is the result of a logical and systematic process of problem-solving. The steps we have taken demonstrate the power of algebraic manipulation in transforming complex problems into simpler, more manageable forms. This final answer is not only the solution to the problem but also a testament to the effectiveness of the algebraic techniques we have employed. It represents a clear and understandable answer that can be used for further analysis or application. The journey from the initial problem to this final answer highlights the beauty and elegance of mathematics in providing solutions to complex problems.

In conclusion, solving the problem ${\frac{x+5}{x+2}-\frac{x+1}{x^2+2x}}$ involved a series of algebraic manipulations, including factoring, finding a common denominator, subtracting numerators, and simplifying the resulting expression. The process demonstrated the importance of these fundamental algebraic techniques in handling rational expressions. Each step was crucial in arriving at the final simplified form, showcasing the interconnectedness of algebraic concepts. The ability to factor, find common denominators, and simplify expressions are essential skills in algebra and are applicable to a wide range of mathematical problems. This exercise not only provided a solution to a specific problem but also reinforced the underlying principles of algebraic manipulation. The journey from the initial problem to the final answer highlighted the power of systematic problem-solving and the importance of attention to detail. The final answer, ${\frac{x^2 + 4x - 1}{x(x+2)}}$, represents the difference between the two original fractions in its most concise and understandable form. This process underscores the elegance and efficiency of algebraic methods in simplifying complex expressions and finding solutions to mathematical problems. The skills acquired through this exercise will be invaluable in tackling more advanced algebraic challenges. The understanding gained from this specific problem extends beyond the immediate solution, providing a foundation for further exploration and mastery of algebraic concepts.