Problem 69 - Algebra for College Students

Question

Find the indicated value for each given rational expression, if possible. $$R(x)=\frac{3 x-5}{x+4}, R(3)$$

Step-by-Step Solution

Answer

The value of \( R(3) \) is \( \frac{4}{7} \).

Step 1: Identify the given rational expression and the value to be found

The rational expression given is \( R(x) = \frac{3x - 5}{x + 4} \). The task is to find the value of this expression when \( x = 3 \).

Step 2: Substitute the value into the rational expression

Substitute \( x = 3 \) into the rational expression \( R(x) \). This gives us \( R(3) = \frac{3(3) - 5}{3 + 4} \).

Step 3: Perform the calculations in the numerator

Calculate the numerator: \( 3(3) - 5 = 9 - 5 = 4 \).

Step 4: Perform the calculations in the denominator

Calculate the denominator: \( 3 + 4 = 7 \).

Step 5: Simplify the expression

Combine the results from the numerator and the denominator. Thus, \( R(3) = \frac{4}{7} \). This is the final value.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. These expressions work much like regular fractions but involve algebraic variables. For example, in the given exercise, the rational expression is \( R(x) = \frac{3x - 5}{x + 4} \). Understanding rational expressions is crucial since they appear frequently in algebra. They are used to describe ratios of polynomial quantities and often show up in real-world applications.

Substitution

Substitution is the process of replacing a variable in an expression with a specific value. This is often the first step in evaluating expressions. In our exercise, we were given the rational expression \( R(x) = \frac{3x - 5}{x + 4} \) and asked to find the value of \( R(3) \). We substitute \( x = 3 \) into the expression: \( R(3) = \frac{3(3) - 5}{3 + 4} \). Substitution simplifies the expression to numeric calculations, making it easier to evaluate.

Simplification

Simplification is essential in mathematics to make expressions as straightforward as possible. For rational expressions, this involves simplifying both the numerator and the denominator. In our exercise, after substitution, we simplify the numerator from \( 3(3) - 5 \) to \( 9 - 5 = 4 \) and the denominator from \( 3 + 4 \) to \( 7 \). Thus, the simplified form of our rational expression is \( \frac{4}{7} \). Simplification helps to obtain a final, cleaner answer.

Numerator and Denominator

In any fraction, including rational expressions, the top part is the numerator and the bottom part is the denominator. Understanding these parts is crucial for operations like addition, subtraction, multiplication, and division of fractions. For the expression \( R(x) = \frac{3x - 5}{x + 4} \), the numerator is \( 3x - 5 \) and the denominator is \( x + 4 \). When substituting \( x = 3 \), we get a new numerator of \( 4 \) and denominator of \( 7 \). Thus, knowing how to handle the numerator and the denominator is key to correctly evaluating rational expressions.