Problem 45 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-2}+\frac{4}{2-x}$$

Step-by-Step Solution

Answer

The simplified result of the given expression is \(x+2\).

Step 1 - Identifying Additives Inverse

In the given problem, identify that the fractions \(\frac{x^{2}}{x-2}\) and \(\frac{4}{2-x}\) have denominators \(x-2\) and \(2-x\), which are additives inverse of each other. This means they are the same but with opposite sign, as \(x - 2 = - (2 - x)\).

Step 2 - Factoring Out -1

You can factor out -1 from the fraction with the denominator \(2-x\). When done, the equation becomes \(\frac{x^{2}}{x-2} - \frac{4}{x-2}\). The fractions now have the same denominator.

Step 3 - Combining the Fractions

Now that the fractions have the same denominator, they can be combined into a single fraction where the numerators are added or subtracted as indicated. The equation can be re-written as \(\frac{x^{2} - 4}{x-2}\).

Step 4 - Simplifying the result

The numerator can be factored into \(x^{2} - 4 = (x+2)(x-2)\). Substituting this into the equation, you get \(\frac{(x+2)(x-2)}{x-2}\). This leaves us with \(x+2\).

Key Concepts

Additive Inverses

When dealing with fractions, especially in algebra, it's important to recognize the concept of additive inverses. Additive inverses are pairs of numbers or terms whose sum is zero. For the given expression \( \frac{x^{2}}{x-2} + \frac{4}{2-x} \), notice that the denominators \( x-2 \) and \( 2-x \) are additive inverses.
To fully understand this, look at the equation \( x-2 = -(2-x) \). This shows that by switching the signs, these denominators can become equal. Recognizing additive inverses is a useful skill in algebra as it helps in simplifying expressions by aligning terms correctly, which makes manipulation easier in subsequent steps.

Simplifying Fractions

Simplifying fractions is all about finding ways to reduce them to their simplest form. In this exercise, simplifying the expression \( \frac{x^{2} - 4}{x-2} \) involves recognizing that the numerator is a difference of squares, \( x^2 - 4 \), that can be factored into \((x+2)(x-2)\).
Upon factoring, the expression becomes \( \frac{(x+2)(x-2)}{x-2} \). This allows us to cancel \( (x-2) \) from the numerator and denominator, since division by a non-zero term is valid.

• Factorization: This technique involves breaking down an expression into simpler terms that multiply together to form the original. It's incredibly useful for simplifying expressions.
• Cancelling like terms allows the fraction to be reduced easily, provided the term isn't zero, preserving the equation's balance.

Simplifying fractions helps clarify expressions and reduce complexity, making it easier to work with them in equations.

Combining Fractions

To combine fractions, especially in algebra, you need a common denominator. In the equation \( \frac{x^{2}}{x-2} + \frac{4}{2-x} \), identifying additive inverses helped us establish a common denominator.
Once the denominators are aligned, the fractions can be combined by adding or subtracting their numerators. After factoring out \(-1\) in the original problem, the expression became \( \frac{x^{2}-4}{x-2} \).

• Having a common denominator simplifies addition or subtraction across fractions.
• Combine numerators after aligning denominators for a single expression representation.

Thus, by combining and simplifying appropriately, the problem simplifies significantly, displaying how crucial a common denominator is in such operations.