Simplify Algebraic Expressions And Rational Functions

This section focuses on simplifying algebraic expressions, particularly those involving fractions with common denominators. Simplifying expressions is a fundamental skill in algebra, essential for solving equations, working with functions, and tackling more advanced mathematical concepts. When dealing with fractions, the process often involves combining like terms and reducing the expression to its simplest form. Understanding how to manipulate these expressions efficiently is crucial for success in algebra and beyond. Let's delve into the specifics of simplifying such expressions with common denominators.

a) Combining Fractions with a Common Denominator

Our first task is to simplify the expression rac{5}{x} + rac{2}{x} + rac{3}{x} = rac{4}{x} for all permissible values of x. This problem highlights the basic principle of adding fractions with common denominators. When fractions share a denominator, we can directly add or subtract their numerators while keeping the denominator the same. This simplifies the process significantly, allowing us to combine multiple fractions into a single, more manageable term. However, it's crucial to remember the restriction on the variable x; since division by zero is undefined, x cannot be equal to zero. This restriction is a critical aspect of working with rational expressions and must always be considered. Keeping this in mind, we can proceed with simplifying the given expression, focusing on combining the numerators and reducing the resulting fraction to its simplest form. The ability to identify and apply such restrictions is a key skill in algebraic manipulation, ensuring the validity of our solutions.

To simplify, we first combine the fractions on the left side of the equation:

${ \frac{5}{x} + \frac{2}{x} + \frac{3}{x} = \frac{5 + 2 + 3}{x} = \frac{10}{x}. }$

Now we have the equation:

${ \frac{10}{x} = \frac{4}{x}. }$

To solve for x, we can multiply both sides of the equation by x, keeping in mind that x cannot be zero:

${ 10 = 4 }$

This statement is clearly false. Therefore, there is no solution for x that satisfies the original equation. This outcome underscores the importance of not only manipulating algebraic expressions correctly but also interpreting the results in the context of the original problem. In this case, the lack of a solution indicates an inconsistency in the equation itself, highlighting a crucial aspect of mathematical problem-solving.

In summary, while we successfully combined the fractions, the resulting equation led to a contradiction, demonstrating that not all algebraic expressions have solutions. This exercise reinforces the need for careful algebraic manipulation and the critical evaluation of results.

b) Simplifying Expressions with Variable Terms in the Numerator

Now, let's focus on simplifying the expression rac{5x-3}{2x} + rac{7}{2x} - rac{3x+1}{2x}. This problem extends the concept of combining fractions with common denominators by including variable terms in the numerators. The approach remains the same: since all fractions share the denominator 2x, we can combine the numerators directly. However, we must pay close attention to the signs, especially when subtracting fractions. Distributing the negative sign correctly is crucial to avoid errors. Additionally, as before, we must consider the permissible values of x. In this case, x cannot be zero, as it would lead to division by zero. This restriction is a fundamental aspect of working with rational expressions and must be consistently applied. By carefully combining the numerators and simplifying the resulting expression, we can reduce the fraction to its simplest form, gaining a deeper understanding of algebraic manipulation.

To simplify this expression, we combine the numerators over the common denominator 2x:

${ \frac{5x - 3}{2x} + \frac{7}{2x} - \frac{3x + 1}{2x} = \frac{(5x - 3) + 7 - (3x + 1)}{2x}. }$

Next, we simplify the numerator by combining like terms:

${ \frac{5x - 3 + 7 - 3x - 1}{2x} = \frac{2x + 3}{2x}. }$

We can leave the expression in this form, or we can split it into two fractions:

${ \frac{2x + 3}{2x} = \frac{2x}{2x} + \frac{3}{2x} = 1 + \frac{3}{2x}. }$

This simplification demonstrates how algebraic expressions can be manipulated into different forms while maintaining their equivalence. The choice of which form to use often depends on the specific context or the desired outcome. The ability to move fluently between different forms of an expression is a valuable skill in algebra.

Therefore, the simplified expression is rac{2x + 3}{2x} or 1 + rac{3}{2x}, with the restriction that x ≠ 0.

c) Simplifying Expressions with Opposite Denominators

Our next challenge is to simplify the expression rac{3x+1}{x-2} + rac{2x-5}{2-x}. This problem introduces a slight twist: the denominators are opposites of each other (x - 2 and 2 - x). To combine these fractions, we need a common denominator. The key is to recognize that we can make the denominators the same by multiplying one of the fractions by -1/-1. This manipulation allows us to add the fractions in a straightforward manner. It's important to carefully distribute the negative sign to avoid errors. Additionally, we must identify the values of x that make the denominator zero, as these values are not permissible. In this case, x cannot be 2. By understanding how to handle opposite denominators and identifying restrictions on the variable, we can successfully simplify this expression and further develop our algebraic skills.

To simplify this expression, we notice that the denominators are opposites. We can rewrite the second fraction by factoring out a -1 from the denominator:

${ \frac{2x - 5}{2 - x} = \frac{2x - 5}{-(x - 2)} = -\frac{2x - 5}{x - 2}. }$

Now we can rewrite the original expression as:

${ \frac{3x + 1}{x - 2} - \frac{2x - 5}{x - 2}. }$

Combine the numerators over the common denominator:

${ \frac{(3x + 1) - (2x - 5)}{x - 2} = \frac{3x + 1 - 2x + 5}{x - 2} = \frac{x + 6}{x - 2}. }$

This simplified expression highlights the importance of recognizing patterns and using algebraic manipulations to create common denominators. By changing the form of the second fraction, we were able to combine the expressions and simplify the result. This ability to manipulate expressions strategically is a cornerstone of algebraic proficiency.

Therefore, the simplified expression is rac{x + 6}{x - 2}, with the restriction that x ≠ 2.

This section transitions into a broader exploration of simplifying algebraic expressions while explicitly identifying any restrictions on the variables. Simplifying expressions is a core skill in algebra, and understanding the restrictions on variables is equally crucial. Restrictions arise when certain values of a variable would lead to undefined operations, such as division by zero or taking the square root of a negative number (in the realm of real numbers). Identifying these restrictions ensures that our simplified expressions are valid for all permissible values of the variable. This section will delve into various techniques for simplifying expressions and the methods for determining and stating these crucial restrictions. Mastering these skills is essential for a thorough understanding of algebraic manipulation.

(The problem statement for question 2 is incomplete. Please provide the expression to be simplified so I can complete this section.)

In summary, simplifying algebraic expressions involves combining like terms, factoring, and applying algebraic identities to reduce an expression to its simplest form. When dealing with rational expressions (fractions with variables in the denominator), it's crucial to identify any values of the variable that would make the denominator zero, as these values are not permissible. By mastering these techniques, we can confidently manipulate algebraic expressions and ensure the validity of our results.