Problem 85 - Algebra A Combined Function

Question

Simplify each rational expression. $$ \frac{x+3}{3+x} $$

Step-by-Step Solution

Answer

The expression simplifies to 1, provided \(x \neq -3\).

Step 1: Recognize the Problem

We are asked to simplify the rational expression \( \frac{x+3}{3+x} \). At first glance, it might seem as if the expression cannot be simplified.

Step 2: Reinterpret the Expression

Observe that \( x + 3 \) and \( 3 + x \) are the same expression, since addition is commutative. Thus, \( x + 3 \) equals \( 3 + x \).

Step 3: Identify Equivalent Expressions

Recognizing that \( x + 3 = 3 + x \), replace the denominator with \( x + 3 \). The expression becomes \( \frac{x+3}{x+3} \).

Step 4: Simplify the Expression

Since \( x+3 \) is equal to the denominator if \( x+3 eq 0 \), \( \frac{x+3}{x+3} \) simplifies to \( 1 \). Note that this is valid only if the denominator is not zero, which means \( x eq -3 \).

Step 5: State the Simplified Form

The rational expression \( \frac{x+3}{3+x} \) simplifies to \( 1 \) for all \( x eq -3 \).

Key Concepts

Simplification

Simplification is a fundamental concept in mathematics, especially when dealing with rational expressions. A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Simplifying a rational expression involves reducing it to its most basic form where no further factoring or reduction is possible.

In the given problem, the rational expression is \( \frac{x+3}{3+x} \). To simplify means to eliminate common factors from the numerator and the denominator so that the expression is as simple as possible. It is important to note that both terms in the numerator and the denominator must match exactly to perform this reduction.

• First, observe whether there is any factor common to both the numerator and the denominator. This deduction often involves reinterpreting and recognizing equivalent expressions, which we'll learn more about in the next sections.
• Once the equivalent nature is identified, the simplification can be applied, as in this exercise where the simplified form becomes \( 1 \).

This process not only helps in deriving more understandable forms of expressions but also aids in solving complex algebraic equations more easily.

Commutative Property

The commutative property is one of the basic properties of addition and multiplication that makes arithmetic readily manageable. It states that numbers can be added or multiplied in any order, and the result will be the same.

In mathematical terms:

• For addition, the commutative property states that \( a + b = b + a \).
• For multiplication, it implies \( ab = ba \).

In our case of simplifying \( \frac{x+3}{3+x} \), the use of the commutative property is shown in the expression's rearrangement. We observe that the numerator \( x + 3 \) and the denominator \( 3 + x \) are the same due to this property.
This observation allows us to claim their equivalency, leading to efficient simplification. Understanding and applying fundamental properties like these can reveal underlying equivalences, streamlining various mathematical processes.

Equivalent Expressions

Equivalent expressions are expressions that, although they may look different at first glance, represent the same value in every context.
In algebra, identifying equivalent expressions is crucial, as it allows for simplification and easier comparison between mathematical formulas.

Using the commutative property, we identified that \( x + 3 \) and \( 3 + x \) are equivalent expressions in this exercise.

• This realization allows us to rewrite the denominator \( 3 + x \) as \( x + 3 \), which is identical to the numerator, thus facilitating further simplification.

Equivalent expressions can emerge from applying not only commutative property but also associative and distributive properties. Recognizing these equivalences is a key skill in algebra, aiding in solving equations and proving identities with ease.

Undefined Values

A crucial aspect to consider when working with rational expressions is the concept of undefined values. These occur in division when the denominator equals zero, since division by zero is undefined in mathematics.

For our simplified expression \( \frac{x+3}{x+3} \) which results in \( 1 \), we must always check what values of \( x \) would make the denominator zero.

• Here, setting \( x+3 = 0 \) gives us \( x = -3 \), pointing out that the expression is undefined at \( x = -3 \).

This means when \( x = -3 \), our original rational expression, as well as the simplified form \( 1 \), does not hold true.
Taking note of undefined values is essential in algebra to ensure the expressions and equations only include meaningful and real values.