Problem 8 - Intermediate Algebra

Question

$$\frac{x+3}{x+3}=\square$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Condition

In the fraction \( \frac{x+3}{x+3} \), the numerator and the denominator are the same expression. This fraction is undefined when the denominator is zero.

Step 2: Determine When the Denominator is Zero

Set the denominator equal to zero: \( x + 3 = 0 \). Solve for \( x \) to find when the fraction is undefined. \( x = -3 \) is when the denominator equals zero, meaning the expression is undefined at this point.

Step 3: Simplify the Fraction for Defined Values

For \( x eq -3 \), the fraction \( \frac{x+3}{x+3} \) simplifies to \( 1 \) because any non-zero number divided by itself is 1.

Key Concepts

Simplifying Algebraic Fractions

Simplifying algebraic fractions is a fundamental skill in algebra that helps make expressions easier to work with. It involves the process of reducing fractions by canceling out common factors in the numerator and the denominator. In the case of the fraction \( \frac{x+3}{x+3} \), the numerator and the denominator are identical.

• This expression can be simplified easily by noting that any non-zero value divided by itself equals 1.
• The simplification is valid for any values except those that make the denominator zero, as division by zero is undefined.

A practical rule of thumb is to first check for any factors that can be canceled. Here, the entire expression \( x+3 \) cancels itself out in both the numerator and denominator, resulting in a simplified form of 1, given that the condition for the denominator being non-zero is satisfied.

Conditions for Undefined Fractions

In the context of algebraic fractions, a fraction becomes undefined whenever its denominator equals zero. Understanding when a fraction is undefined is crucial in solving equations and performing algebraic manipulations.
For the given exercise \( \frac{x+3}{x+3} \), determining when the expression is undefined involves identifying when the denominator, \( x+3 \), turns to zero:

• Set the denominator equal to zero: \( x+3 = 0 \).
• Solve for \( x \) to find the critical value.
• The expression becomes undefined specifically at \( x = -3 \).

By pinpointing the value of \( x \) that causes the fraction to lose its definition, you are able to understand the restrictions through which the algebraic expression operates.

Solving Equations for Restrictions

When dealing with algebraic fractions, it’s important to solve equations to establish any restrictions that may prevent simplification or cause the expression to become undefined. These restrictions often hinge on values that make a denominator zero.
For our exercise, the primary concern is to avoid division by zero, a fundamental rule:

• The denominator, in this case, \( x+3 \), must not equal zero, so calculate \( x \) by setting \( x+3=0 \).
• Solving gives \( x=-3 \).
• This forms a condition: \( x eq -3 \).

Identifying these conditions allows you to simplify the expression correctly and avoids errors in calculations. Always check for restrictions before performing any operations on algebraic fractions to ensure that the solution remains valid across the problem's intended domain.