Problem 45 - Elementary Algebra

Question

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{x+3}+\frac{2}{x-2}}{\frac{2}{x+3}-\frac{3}{x-2}} $$

Step-by-Step Solution

Answer

\( \frac{3x+4}{-x-13} \)

Step 1: Identify the common denominators

The expression contains two fractions in both the numerator and the denominator. Identify the common denominators for each part. For the numerator, the common denominator is \((x+3)(x-2)\). Similarly, for the denominator, the common denominator is also \((x+3)(x-2)\).

Step 2: Rewrite each fraction with the common denominator

Rewrite each fraction with the identified common denominator. The numerator becomes: \[ \frac{1(x-2)+2(x+3)}{(x+3)(x-2)} = \frac{x-2+2x+6}{(x+3)(x-2)} = \frac{3x+4}{(x+3)(x-2)} \] Likewise, the denominator becomes: \[ \frac{2(x-2)-3(x+3)}{(x+3)(x-2)} = \frac{2x-4-3x-9}{(x+3)(x-2)} = \frac{-x-13}{(x+3)(x-2)} \]

Step 3: Simplify the quotients

Now, express the original expression as: \[ \frac{\frac{3x+4}{(x+3)(x-2)}}{\frac{-x-13}{(x+3)(x-2)}} \] This simplifies to: \[ \frac{3x+4}{(x+3)(x-2)} \times \frac{(x+3)(x-2)}{-x-13} = \frac{3x+4}{-x-13} \]

Step 4: Simplify the final expression

The common factors \((x+3)(x-2)\) cancel out. Therefore, the simplified form of the original expression is: \[ \frac{3x+4}{-x-13} \]

Key Concepts

Common Denominator

In algebra, fractions often need to share a common base to be added or subtracted easily. The common denominator is the same value in the denominator for both fractions. This allows us to combine fractions smoothly.

In our problem, we had two fractions: \(\frac{1}{x+3}\frac{2}{x-2}\). To add these fractions, we need a common denominator. We can find this by multiplying the denominators together, so our common denominator is \((x+3)(x-2)\).

Always remember to apply the common denominator to both the numerator and the denominator fractions when simplifying or evaluating algebraic fractions.

Simplifying Expressions

Simplifying an algebraic expression means making it as straightforward as possible. This often involves combining like terms or reducing fractions.

In our exercise, we simplified \(\frac{3x+4}{(x+3)(x-2)}\) and \(\frac{-x-13}{(x+3)(x-2)}\) by canceling the common denominator, \((x+3)(x-2)\). This left us with \(\frac{3x+4}{-x-13}\).

Every simplifying step aims to make the fractions or expressions easier to work with, and it can involve combining terms or cancelling out similar components.

Fraction Multiplication

Fraction multiplication is more straightforward than addition or subtraction. You simply multiply the numerators together and the denominators together.

After combining the fractions with a common denominator, we needed to multiply them to simplify the quotient: \(\frac{\frac{3x+4}{(x+3)(x-2)}}{\frac{-x-13}{(x+3)(x-2)}}\). This simplifies by multiplying the numerator of the first fraction by the denominator of the second: \(\frac{3x+4}{(x+3)(x-2)}\ \times \frac{(x+3)(x-2)}{-x-13}\). The \((x+3)(x-2)\) terms cancel, leaving the simplified fraction \(\frac{3x+4}{-x-13}\).

This operation highlights the clean-up power of multiplication in fraction algebra.

Numerator and Denominator

A fraction consists of two key parts: the numerator and the denominator. The numerator is the top part, representing how many parts are concerned, while the denominator at the bottom defines the total number of parts.

Understanding this is crucial when dealing with algebraic fractions. For instance, in the problem, our initial numerator fractions were \(\frac{1}{x+3}+ \frac{2}{x-2}\) and our denominator fractions were \(\frac{2}{x+3}-\frac{3}{x-2}\).

We rewrote them over a common denominator to make the parts easier to compare and simplify: \(\frac{3x+4}{(x+3)(x-2)}\) for the numerator and \(\frac{-x-13}{(x+3)(x-2)}\) for the denominator. Recognizing and manipulating these pieces correctly makes tasks like addition, subtraction, and simplification achievable.