Problem 63 - College Algebra

Question

Simplify each complex rational expression. $$ \frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}} $$

Step-by-Step Solution

Answer

-\frac{x}{7} + 2

Step 1: Combine the Numerator Expressions

To combine the expressions in the numerator, find a common denominator which in this case is \((x-2)(x+2)\), multiply each fraction by the missing term in its denominator and then subtract: \[\frac{3}{x-2}-\frac{4}{x+2} = \frac{3(x+2)}{(x-2)(x+2)} - \frac{4(x-2)}{(x-2)(x+2)} = \frac{3x+6 - 4x+8}{(x-2)(x+2)} = \frac{-x+14}{(x-2)(x+2)}\]

Step 2: Simplify the Denominator Expression

Simplify the expression \(\frac{7}{x^{2}-4}\), where \(x^{2}-4\) can be expressed as \((x-2)(x+2)\), so it simplifies to \(\frac{7}{(x-2)(x+2)}\)

Step 3: Invert and Multiply

Invert the second fraction and multiply both complex fractions: \[\frac{-x+14}{(x-2)(x+2)} * \frac{(x-2)(x+2)}{7} = \frac{-x+14}{7}\]

Step 4: Final Simplification

Simplify the resulting expression to get the final result as \(-\frac{x}{7} + 2\).

Key Concepts

Common Denominator

In algebra, when dealing with operations involving fractions, finding a common denominator is essential.This is the number that allows you to combine fractions with different denominators into a single fraction easily.
For the given expression \(\frac{\frac{3}{x-2}-\frac{4}{x+2}}{\frac{7}{x^{2}-4}}\), the goal is to combine the fractions in the numerator.

• Identify the denominators: \(x-2\) and \(x+2\).
• The least common multiple of these expressions is \((x-2)(x+2)\), which is also known as the common denominator.

To combine the fractions, adjust each fraction to have this common denominator by multiplying:

• First fraction: \(\frac{3}{x-2} \times \frac{x+2}{x+2} = \frac{3(x+2)}{(x-2)(x+2)}\)
• Second fraction: \(\frac{4}{x+2} \times \frac{x-2}{x-2} = \frac{4(x-2)}{(x-2)(x+2)}\)

Once the denominators match, the two fractions can be subtracted easily, completing the first step of the simplification.

Invert and Multiply

"Invert and multiply" is a common method used to divide fractions.After simplifying the individual fractions, apply this method to complex rational expressions as well.
In the example \(\frac{\frac{-x+14}{(x-2)(x+2)}}{\frac{7}{(x-2)(x+2)}}\), to simplify, you flip the second fraction and multiply instead of dividing:

• The fraction \(\frac{7}{(x-2)(x+2)}\) becomes \(\frac{(x-2)(x+2)}{7}\) when inverted.
• Multiply: \(\frac{-x+14}{(x-2)(x+2)} \times \frac{(x-2)(x+2)}{7} = \frac{-x+14}{7}\).

This step transforms a division into a multiplication problem, which is often simpler to compute and visualize.

Algebraic Simplification

Simplification in algebra involves turning complex expressions into simpler ones without changing their value. After using arithmetic operations and applying fraction rules, you arrive at a simplified form.
In the final step of our example, focus on simplifying the expression obtained from the "invert and multiply" step: \(\frac{-x+14}{7}\).

• The numerator \(-x+14\) can be separated and expressed as \(-\frac{x}{7} + \frac{14}{7}\).
• This further simplifies to \(-\frac{x}{7} + 2\), ensuring you have the simplest possible form of the original complex rational expression.

Remember, proper algebraic simplification makes an expression easier to understand and further operate with if needed.