Problem 109 - Introductory Algebra for College Students

Question

Use the GRAPH or TABLE feature of a graphing utility to determine if the rational expression has been correctly simplified. If the simplification is wrong, correct it and then verify your answer using the graphing utility. $$\frac{3 x+15}{x+5}=3, x \neq-5$$

Step-by-Step Solution

Answer

The given rational expression \( \frac{3x+15}{x+5} \) has been correctly simplified to 3, for all x not equal to -5.

Step 1: Analyze the Simplification

The rational expression \( \frac{3x+15}{x+5} \) is simplified to 3. To validate this, isolate each part of the expression. Start by observing the numerator, which is '3x+15'. This can be simplified by pulling out a factor of 3 to get: 3(x+5). The denominator is 'x+5'. After pulling out a factor of 3 from the numerator, it looks identical to the denominator. Hence, the simplified form is 3.

Step 2: Check condition

The condition \( x \neq -5 \) is to avoid division by zero which is undefined in mathematical context. Hence, the simplification holds true for all values of x, except -5.

Step 3: Verification using Graphing Utility

To verify, plot the given rational expression and the simplified form in a graphing utility. The graph of the rational expression \( \frac{3x+15}{x+5} \) should overlap with the graph of the constant 3, except at \( x = -5 \), because of the condition \( x \neq -5 \). The function will be undefined at \( x = -5 \). This confirms the correctness of the simplification.

Key Concepts

Graphing Utility Verification

Graphing utilities are invaluable tools for verifying the simplification of rational expressions. They provide a visual representation that can confirm whether the simplified expression and the original expression are equivalent. When using a graphing utility, it's important to input both the original rational expression and its simplified form. The graphs should overlay each other closely, differing only at values which are excluded from the domain, such as points where the original expression would cause division by zero.

For the example given, a graph of the expression \( \frac{3x+15}{x+5} \) and the number 3 should coincide. However, at \( x = -5 \), the graph of the rational expression will not be defined, indicating that despite the simplification, the two expressions are not identical in their domains. This visual check confirms the solution and helps to solidify the understanding of the simplification process.

Rational Expression Simplification

Simplifying rational expressions involves reducing them to their simplest form. This is achieved by factoring both the numerator and the denominator and cancelling any common factors. For clarity, let's take a closer look at the given example. We start with the rational expression \( \frac{3x+15}{x+5} \). Factor out the greatest common divisor in the numerator, which is 3, to obtain \( 3(x+5) \). The denominator remains \( x+5 \). The common factors in the numerator and denominator \( (x+5) \) cancel out, leaving us with the simplified expression 3.

While this process may seem straightforward, careful attention must be paid to any possible restrictions in the domain to ensure accuracy and avoid mathematical errors.

Conditions for Simplification

Simplification of expressions has to respect certain conditions to ensure the equivalence of the original and simplified forms. Let's discuss the vital conditions to be aware of:

• Finding and cancelling common factors correctly.
• Keeping track of any non-permissible values for the variable which might lead to division by zero.
• Ensuring that any simplifications do not alter the original domain of the expression.

In the context of the example, the condition \( x eq -5 \) ensures the original domain is preserved post-simplification. Ignoring this condition could lead to incorrect interpretations, particularly when graphing or applying the expression to real-world scenarios.

Avoiding Division by Zero

In mathematics, division by zero is undefined, and it's crucial to avoid it when working with rational expressions. Whenever you simplify a rational expression, it is imperative to identify any values of the variable that would make the denominator zero and exclude those from the domain.

Why is this significant?

Ensuring that the domain excludes points where the denominator is zero maintains the integrity of the expression function and prevents mathematical fallacies.

For the provided exercise, the exclusion of \( x = -5 \) is notable. At this point, the original denominator \( x+5 \) is zero, leading to an undefined expression. Recognizing and applying this condition wards off potential errors and accurately reflects the behavior of the rational expression across its entire domain.