The Timpson Lyceum · Intermediate Algebra (2023)

Lecture 15 (Intermediate Algebra) Rational Expressions

Overview

This lecture finishes the work with rational expressions by tackling complex fractions — fractions whose numerator and/or denominator contain fractions themselves — and then applies rational expressions to solving equations. The strategy for a complex fraction mirrors ordinary fraction arithmetic: combine the numerator into a single fraction, combine the denominator into a single fraction, then divide by multiplying by the reciprocal. From there the lecture moves into solving proportions by cross-multiplication, clearing fractions from rational equations by multiplying through by the least common denominator (LCD), and finally unit conversion using the factor-label method. Mastering these tools lets you simplify intimidating stacked expressions and solve equations that contain variables in denominators.

Key Concepts

Complex fraction — A fraction that has one or more fractions in its numerator, its denominator, or both. To simplify:

Combine the numerator into a single fraction (find a common denominator).
Combine the denominator into a single fraction the same way.
Divide the two: multiply the top fraction by the reciprocal of the bottom fraction.
Simplify/cancel, and factor when possible (a difference of squares often hides in the result).

Dividing fractions — For any fractions, $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b}\cdot\dfrac{d}{c}$ . "Invert and multiply."

Difference of squares — Always factorable: $x^2 - 1 = (x-1)(x+1)$ , and more generally $a^2 - b^2 = (a-b)(a+b)$ . Watch for it after simplifying a complex fraction so the result reduces fully.

Negative exponents — A negative exponent means take the reciprocal: $m^{-2} = \dfrac{1}{m^2}$ . Rewrite every negative-exponent term as a positive-exponent fraction first, then treat the expression as an ordinary complex fraction. The relocated factor goes into the denominator by itself — do not merge it into a neighboring term's denominator.

Cross-multiplication (proportions) — If $\dfrac{a}{b} = \dfrac{c}{d}$ , then $ad = bc$ . This follows from multiplying both sides by $b$ and by $d$ . It turns a proportion into a polynomial equation you already know how to solve.

If cross-multiplying produces a squared term, move everything to one side, set equal to $0$ , and solve by factoring.
Always check solutions by substituting back into the original proportion.

Clearing fractions from an equation — Multiply both sides by the LCD of all denominators. Each denominator cancels, leaving a fraction-free equation. If a variable expression like $x+2$ sits in a denominator, it becomes part of the LCD.

Proportional reasoning (word problems) — Quantities that scale together keep a constant ratio. For similar shadows, $\dfrac{\text{height}}{\text{shadow}}$ is the same for every object, so set the two ratios equal and cross-multiply.

Unit conversion (factor-label method) — Multiply by conversion factors equal to $1$ (e.g. $\dfrac{5280\text{ ft}}{1\text{ mi}}$ ) so that unwanted units cancel and the desired units remain. For squared or cubed units, cube the linear factor: $1\text{ yd} = 3\text{ ft}$ but $1\text{ yd}^3 = 3^3 = 27\text{ ft}^3$ .

Worked Example

Simplify the complex fraction $\dfrac{\,x - \dfrac{1}{x}\,}{\,1 - \dfrac{1}{x}\,}$ .

Combine numerator and denominator separately, then multiply by the reciprocal and factor:

\begin{aligned} \frac{x - \dfrac{1}{x}}{\,1 - \dfrac{1}{x}\,} &= \frac{\dfrac{x^2 - 1}{x}}{\dfrac{x - 1}{x}} \\ &= \frac{x^2 - 1}{x}\cdot\frac{x}{x - 1} \\ &= \frac{(x-1)(x+1)}{x}\cdot\frac{x}{x - 1} \\ &= x + 1 \end{aligned}

Solve the proportion $\dfrac{k+3}{6} = \dfrac{8}{k-2}$ by cross-multiplying, then factoring:

\begin{aligned} (k+3)(k-2) &= 24 \\ k^2 + k - 6 &= 24 \\ k^2 + k - 30 &= 0 \\ (k+6)(k-5) &= 0 \\ k &= -6 \quad\text{or}\quad k = 5 \end{aligned}

Summary

Simplify a complex fraction by combining the top into one fraction, the bottom into one fraction, then multiplying the top by the reciprocal of the bottom.
After simplifying, look for a difference of squares and factor so the expression reduces fully.
Rewrite negative exponents as positive-exponent fractions before simplifying; the relocated factor stands alone in its denominator.
A proportion $\dfrac{a}{b} = \dfrac{c}{d}$ becomes $ad = bc$ by cross-multiplication; if a square appears, set the equation to $0$ and factor.
Clear fractions from any rational equation by multiplying both sides by the LCD, then solve the resulting polynomial.
Convert units with factor-label ratios equal to $1$ so units cancel; for area/volume, raise the linear conversion to the 2nd/3rd power ( $1\text{ yd}^3 = 27\text{ ft}^3$ ).