Problem 63 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{4 x}{x-5}-3 $$

Step-by-Step Solution

Answer

\( \frac{x+15}{x-5} \)

Step 1: Understand the Problem

We need to subtract the constant 3 from the rational expression \( \frac{4x}{x-5} \). This involves rewriting 3 in terms of the same denominator as \( \frac{4x}{x-5} \).

Step 2: Rewrite the Constant as a Rational Expression

To subtract 3 from \( \frac{4x}{x-5} \), rewrite 3 with a denominator of \( x-5 \). Multiply both the numerator and denominator of 3 by \( x-5 \) to get \( \frac{3(x-5)}{x-5} \).

Step 3: Perform the Subtraction

Now we can subtract \( \frac{3(x-5)}{x-5} \) from \( \frac{4x}{x-5} \). The expression becomes: \[\frac{4x}{x-5} - \frac{3(x-5)}{x-5} = \frac{4x - 3(x-5)}{x-5}.\]

Step 4: Simplify the Numerator

Distribute the -3 in the expression \( 4x - 3(x-5) \): \[4x - 3x + 15 = x + 15.\]

Step 5: Write the Final Expression in Simplest Form

The simplified form of the original problem is: \[\frac{x+15}{x-5}.\] There are no common factors in the numerator and denominator, so this is the simplest form.

Key Concepts

Addition and Subtraction of Rational Expressions

When tackling rational expressions, one essential skill is addition and subtraction. These processes are much like working with regular fractions. But here, you deal with algebraic expressions. To add or subtract rational expressions, do the following:

• Find a Common Denominator: Just like with fractions, you need a common denominator. Check both expressions for a common term. If they don't share one, you have to multiply denominators with the necessary adjustments to make them the same.
• Adjust the Numerators: If you change the denominator, remember to adjust the numerator by the same factor. This keeps the expressions equivalent.
• Add or Subtract the Numerators: Once they have a common denominator, combine the numerators. Keep the denominator the same.
• Simplify: Always simplify the result if possible. Look for common factors in the new numerator and denominator. Reduce them if they exist.

For example, in our original exercise, we needed to subtract 3 from \( \frac{4x}{x-5} \). This involved converting 3 into \( \frac{3(x-5)}{x-5} \) before doing the subtraction. This way, the operation was straightforward, as both expressions shared the same denominator.By following these steps, you can confidently handle any addition or subtraction problem involving rational expressions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is all about making them as clear and concise as possible. This often involves removing unnecessary components or reducing expressions to their simplest forms. Here's a process to follow:

• Combine Like Terms: Start by identifying any terms that are alike. These are terms that have the same variable raised to the same power. Combine them into a single term.
• Factor Where You Can: Check if the expression - particularly the numerator and denominator - can be factored. This sometimes reveals common factors.
• Cancel Out Common Factors: If the numerator and denominator share common factors, cancel them out. This simplifies the expression and makes it clearer.

Simplification helps make expressions easier to work with and understand. In the original exercise, the expression was \( \frac{4x - 3(x-5)}{x-5} \). Simplification involved expanding and combining like terms in the numerator, yielding \( x + 15 \). Since the numerator and denominator had no common factors, this was the simplest form.

Algebraic Fractions

An algebraic fraction is an expression that contains a polynomial in the numerator and denominator. Working with these expressions can seem tricky, but they follow the same basic rules as numerical fractions.

• Understand the Components: The numerator and the denominator can both be polynomials. This is what distinguishes algebraic fractions from numerical fractions.
• Simplify When Possible: Simplifying an algebraic fraction involves factoring both the numerator and the denominator. If they share common factors, you can simplify by canceling them out.
• Be Cautious of Restrictions: Remember that the denominator cannot be zero. So, always determine the values of the variable(s) that would make the denominator zero and exclude them from the solution.

In our example, the expression \( \frac{4x}{x-5} \) involves a polynomial numerator and denominator. When working with it, be mindful that \( x eq 5 \), or the expression becomes undefined. As you practice and grow familiar with algebraic fractions, the processes of simplifying and performing operations with them become more intuitive.