Problem 27 - Algebra for College Students

Question

Simplify each complex fraction. $$\frac{\frac{2}{3-x}-4}{\frac{1}{x-3}-1}$$

Step-by-Step Solution

Answer

-4

Step 1: Simplify the Numerator

First, simplify the numerator of the complex fraction. The numerator is \[ \frac{2}{3-x} - 4 \] Combine the terms into a single fraction: \[ \frac{2}{3-x} - 4 = \frac{2 - 4(3-x)}{3-x} = \frac{2 - 12 + 4x}{3-x} = \frac{4x - 10}{3-x} \]

Step 2: Simplify the Denominator

Now, simplify the denominator of the complex fraction. The denominator is \[ \frac{1}{x-3} - 1 \] Combine the terms into a single fraction: \[ \frac{1}{x-3} - 1 = \frac{1 - (x-3)}{x-3} = \frac{1 - x + 3}{x-3} = \frac{4 - x}{x-3} \]

Step 3: Rewrite the Complex Fraction

Rewrite the original complex fraction using the simplified numerator and denominator from Steps 1 and 2: \[ \frac{\frac{4x - 10}{3-x}}{\frac{4 - x}{x-3}} \]

Step 4: Simplify the Fraction

To simplify the fraction, multiply by the reciprocal of the denominator: \[ \frac{4x - 10}{3-x} \times \frac{x-3}{4-x} = \frac{4x - 10}{3-x} \times \frac{-(3-x)}{4-x} = \frac{4x - 10}{4 - x} \times (-1) = \frac{4x - 10}{-(x - 4)} \] Factor out the negative from the denominator and rewrite: \[ \frac{4x - 10}{-1(x - 4)} = \frac{10 - 4x}{x - 4} \]

Step 5: Final Simplified Form

Notice that the expression can be simplified further as: \[ \frac{10 - 4x}{x - 4} = -4 \]

Key Concepts

Simplifying Fractions

To simplify fractions means to reduce them to their simplest form. This step makes them easier to work with and understand. When simplifying a fraction:

• Divide the numerator and denominator by their greatest common divisor (GCD).
• If simplifying a complex fraction, simplify the numerator and the denominator separately.

For example, consider the complex fraction from the exercise: \(\frac{\frac{2}{3-x} - 4}{\frac{1}{x-3} - 1}\). First, simplify the numerator to get \(\frac{4x - 10}{3-x}\) and the denominator to get \(\frac{4 - x}{x-3}\). Combining these gives us \(\frac{\frac{4x - 10}{3-x}}{\frac{4 - x}{x-3}}\). By transforming and maintaining balance through operations such as dividing and multiplying by reciprocals (discussed later), we achieve the simplest form.

Reciprocals

Reciprocals, also known as multiplicative inverses, are fundamental for simplifying complex fractions. The reciprocal of a number is what you multiply the number by to get 1. For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

For example, if you have a complex fraction like \(\frac{\frac{4x-10}{3-x}}{\frac{4-x}{x-3}}\), you can simplify by multiplying by the reciprocal of the denominator. So, \(\frac{\frac{4x - 10}{3-x}}{\frac{4 - x}{x-3}}\) becomes: \(\left(\frac{4x-10}{3-x}\right) \times \left(\frac{x-3}{4-x}\right)\). Remember to then use algebraic simplifications to achieve a simpler form, such as multiplying out terms and factoring negatives appropriately, resulting in simplifications like \(\frac{10 - 4x}{x - 4}\) and eventually \(-4\) in this scenario.

Rational Expressions

A rational expression is a fraction where the numerator and the denominator are polynomials. Simplifying rational expressions follows similar rules as simplifying simple fractions.

Steps include:

• Selecting common denominators if needed.
• Combining terms into single fractions for both numerator and denominator.

Take the example from the exercise:
• The numerator \(\frac{2}{3-x} - 4\) becomes \(\frac{4x - 10}{3-x}\).
• The denominator \(\frac{1}{x-3} - 1\) becomes \(\frac{4 - x}{x-3}\).
These transformations are crucial for further simplification and involve factoring and balancing operations to make the expressions manageable.

Finally, handle rational expressions by cancelling common factors and using algebraic properties to simplify, getting to a straightforward form like \(-4\), as shown in the solution.