Problem 46 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-3}+\frac{9}{3-x}$$

Step-by-Step Solution

Answer

The simplified expression is \(x + 3\).

Step 1: Recognize Opposite Denominators

The denominators \(x-3\) and \(3-x\) are opposites. We can confirm this because \(3-x\) is equal to \(-(x-3)\). Therefore, we can add the two fractions directly. This can be done by taking \(x-3\) as the common denominator.

Step 2: Combine the Fractions

The fractions can be combined as follows: \(\frac{x^{2}}{x-3} - \frac{9}{x-3}\), which is because we treat \(3-x\) as -\((x-3)\).

Step 3: Simplify the Expression

The expression now simplifies to \(\frac{x^{2} - 9}{x - 3}\). Now, simplify the numerator using difference of squares to get \(\frac{(x-3)(x+3)}{x-3}\).

Step 4: Simplify the Final Expression

We can now cancel out \((x - 3)\) in the numerator and the denominator to get the final simplified expression, that is \(x + 3\).

Key Concepts

Opposite Denominators

In algebra, recognizing opposite denominators, also known as additive inverses, plays an important role in simplifying rational expressions. Two expressions are opposite if one is the negative of the other.
For example, noticing that the denominators \(x-3\) and \(3-x\) are opposites is crucial. This means \(3-x = -(x-3)\). This property allows us to change the form of a fraction for easy addition or subtraction.
When you encounter opposite denominators:

• Understand that they are reflections of each other.
• Recognize they can be unified by accounting for the negative sign.
• Use one of the forms, commonly \(x-3\) in this scenario, as the denominator for both terms.

By using the common form, calculations become more straightforward, enabling you to work towards simplifying the expressions.

Difference of Squares

The difference of squares is a formula used to simplify expressions and equations. It plays a pivotal role when working with rational expressions. A difference of squares looks like \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\).
In the exercise, we encounter \(x^2 - 9\) in the numerator, which is a difference of squares:

• Recognize \(x^2 - 9\) is \((x-3)(x+3)\) because \(9\) is the square of \(3\).
• This happens because \(x^2\) is the square of \(x\), and \(9\) is the square of \(3\).
• Apply the difference of squares to transform \(x^2 - 9\) into \((x-3)(x+3)\), preparing the expression for further simplification.

Recognizing and applying this formula effectively simplifies complex expressions into manageable factored forms, making further computations easier.

Simplifying Expressions

Simplifying expressions, especially complex fractions, is a key aspect of algebra to bring equations to their simplest and most manageable form. After recognizing opposites and identifying a difference of squares, the next step involves actual simplification.
Here’s how you simplify expressions using the example given:

• Once the denominator is unified, the combined fractions become \(\frac{x^2 - 9}{x-3}\).
• Apply the difference of squares to get \(\frac{(x-3)(x+3)}{x-3}\).
• You can then cancel out the common terms \( (x-3) \) found in both the numerator and the denominator.
• This leaves you with \(x+3\) as the final simplified expression.

Identifying opportunities to cancel common factors is essential for simplification, as it reduces expressions to simpler forms and solutions.