Week 11 · Lecture outline

Week 11 — Lecture Outline · Rational Expressions & Equations

College Algebra · MATH 120 Fall 2026 · Prof. Calloway Fictional sample

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective covered: Objective 7 — Simplify, operate on, and solve rational expressions and equations, identifying extraneous solutions.
SLOs touched: A (apply procedures accurately) · B (connect symbolic and numerical representations)
Meeting pattern: 2 sessions × 75 min = 150 min. Segment minutes below total ~150; scale to your own pattern.
Holiday note: Veterans Day (Wed Nov 11) is a campus holiday. Sessions are Tue Nov 10 and Thu Nov 12.

Week at a Glance


The week's big question	"A fraction is a fraction — so why do so many rational-expression mistakes come from trying to cancel things that aren't factors?"
By the end of the week, students can…	(1) simplify a rational expression by factoring numerator and denominator and canceling common factors (not terms); (2) multiply and divide rational expressions by factoring, flipping, canceling; (3) add and subtract with an LCD built from factored denominators; (4) solve a rational equation by clearing the LCD and checking for extraneous solutions.
Key vocabulary	rational expression; excluded value; domain restriction; least common denominator (LCD); simplify; extraneous solution; work-rate problem
Materials	slides (Deck 11); the week's readings + videos; Desmos (or GeoGebra); an approved chatbot (Gemini / Claude / ChatGPT) for the AI-critique moment and the tutorial
Timing note	8 segments, ~150 min total. Session 1 = Segments 1–4 (~73 min). Session 2 = Segments 5–8 (~77 min).

Segment 1 — Hook & the Promise (8 min) · Session 1 opens

Hook — the "obvious" simplification that isn't. Write on the board:

(x + 3) / x = ?

Ask for a show of hands: who would cancel the x to get 3? Pause — usually several hands go up, or nobody raises their hand but you can see who's tempted.

"That cancellation is wrong — and it's the single most common rational-expression error in College Algebra. The x in the numerator is a term — something you add — not a factor — something you multiply. You can only cancel factors."
"But here's the thing: the mistake doesn't come from not understanding fractions. It comes from pattern-matching: you see the same letter on top and bottom, and your brain says 'cancel.' This week we'll replace that reflex with a rule."

The promise (write it): "By Friday you'll simplify, combine, and solve rational expressions without falling into the cancellation trap — and you'll always check your answers for the ghost solutions that rational equations love to produce."

Why it matters: rational expressions show up in rate problems (speed, work, flow), in asymptote analysis (Week 10), and in calculus limits. Getting them right now pays dividends for the rest of the course and beyond.

Segment 2 — Simplifying Rational Expressions (20 min)

Plain language first. A rational expression is a fraction whose numerator and denominator are polynomials. It is undefined (excluded from the domain) wherever the denominator is zero.

Two-step method — always:
1. Factor the numerator and denominator completely.
2. Cancel common factors (expressions that multiply every part of the numerator AND sit as a factor in the denominator).

One fully worked example — every step, every algebra move shown:

Simplify (x² − 9) / (x + 3).

Step 1 — factor the numerator: x² − 9 = (x + 3)(x − 3). (Difference of squares.)

Step 2 — factor the denominator: (x + 3) is already factored.

Step 3 — cancel: (x + 3)(x − 3) / (x + 3) = (x − 3), for x ≠ −3.

Note the domain restriction: x = −3 makes the original denominator zero, so it is excluded even from the simplified form.

One more worked example (quadratic denominator):

Simplify (x² − 4) / (x² + 4x + 4).

Numerator: x² − 4 = (x − 2)(x + 2).

Denominator: x² + 4x + 4 = (x + 2)².

Cancel one factor of (x + 2): = (x − 2) / (x + 2), for x ≠ −2.

The cardinal sin — name it explicitly (memory hook):

❌ (x + 3) / x = 3. WRONG. The x in the numerator is a term (added), not a factor (multiplied). You cannot cancel it.

✅ (3x + 9) / (x + 3) = 3(x + 3) / (x + 3) = 3, x ≠ −3. This is fine — after factoring, (x + 3) is a factor.

The rule that never fails: factor first, then cancel — never cancel until you have factored.

Segment 3 — Multiplying and Dividing Rational Expressions (20 min)

Plain language first. Multiplying rational expressions works exactly like multiplying numerical fractions: multiply numerators, multiply denominators. But factor before you multiply so you can cancel early and keep the numbers small.

Multiplication — fully worked example:

Simplify [(x² − 9) / (x + 2)] · [(x + 2) / (x − 3)].

Factor everything first:
= [(x + 3)(x − 3) / (x + 2)] · [(x + 2) / (x − 3)]

Cancel before multiplying:
(x + 2) cancels across the product; (x − 3) cancels across the product.

Result: (x + 3), for x ≠ −2, x ≠ 3.

Division — flip and multiply:

Divide [(x² − 1) / (x + 3)] ÷ [(x − 1) / (x + 3)].

Rewrite as multiplication: [(x² − 1) / (x + 3)] · [(x + 3) / (x − 1)]

Factor: [(x − 1)(x + 1) / (x + 3)] · [(x + 3) / (x − 1)]

Cancel (x − 1) and (x + 3): = (x + 1), for x ≠ ±1, x ≠ −3.

Named misconception — "I can only cancel from the same fraction":

❌ Students think you must cancel within one fraction before multiplying.
✅ In a product of fractions, cancellation can happen across the product (a factor in one numerator against the same factor in any denominator). That's exactly the rule: A/B · C/D — any factor in {A, C} cancels any matching factor in {B, D}.

Segment 4 — Adding and Subtracting with an LCD (25 min) · Session 1 closes (~73 min)

Plain language first. Adding rational expressions requires a common denominator — exactly like adding numerical fractions. The trick is that the LCD is built from factored polynomial denominators.

LCD recipe (three steps):
1. Factor each denominator completely.
2. List all distinct factors, each to its highest power appearing in any denominator.
3. Multiply those together — that's the LCD.

Fully worked example (simple LCD):

Add 2/x + 3/x.

LCD = x. Both fractions already have the LCD.

2/x + 3/x = (2 + 3)/x = 5/x.

Fully worked example (different denominators — the main technique):

Add 1/x + 1/(x + 1).

Factor denominators: x and (x + 1) — both already factored, no common factors.

LCD: x(x + 1).

Build equivalent fractions:
1/x = (x + 1) / [x(x + 1)]
1/(x + 1) = x / [x(x + 1)]

Add numerators:
(x + 1 + x) / [x(x + 1)] = (2x + 1) / [x(x + 1)].

Subtraction example (sign trap):

Subtract x/(x² − 4) − 1/(x − 2).

Factor: x² − 4 = (x − 2)(x + 2). So LCD = (x − 2)(x + 2).

First fraction already has LCD.

Second fraction: 1/(x − 2) = (x + 2) / [(x − 2)(x + 2)].

Subtract:
[x − (x + 2)] / [(x − 2)(x + 2)]
= [x − x − 2] / [(x − 2)(x + 2)]
= −2 / [(x − 2)(x + 2)].

Misconception + cure:

❌ "I'll just multiply the two denominators to get the LCD." That works only when the denominators share no common factors. If they do, the LCD is smaller than the product — and using the product creates a mess that requires extra simplification.
✅ Factor first, then the LCD writes itself.

Sign error alert — distributing a negative in subtraction:

In A/(…) − B/(…), when you write the combined numerator, A − B — every term in B gets the negative sign, not just the first term. This is the subtract-then-simplify trap, and it costs points every time.

Segment 5 — Solving Rational Equations (20 min) · Session 2 opens

Hook back in: "We've simplified, multiplied, and added. Now we solve. The technique: multiply every term by the LCD to clear all denominators — but that move creates a new danger that rational equations are famous for."

The technique — four steps:
1. Find the excluded values (set each denominator to zero, solve — these values can NEVER be solutions).
2. Find the LCD of all rational terms.
3. Multiply every term in the equation by the LCD (both sides). Simplify — the fractions disappear.
4. Solve the resulting equation (linear or quadratic). Check every solution against the excluded values.

Fully worked example (linear result):

Solve x/3 + 1/2 = 5/6.

Excluded values: none (no variable in any denominator).

LCD: 6.

Multiply through by 6: 6·(x/3) + 6·(1/2) = 6·(5/6)
→ 2x + 3 = 5
→ 2x = 2
→ x = 1.

Check: 1/3 + 1/2 = 2/6 + 3/6 = 5/6. ✓

Fully worked example (variable in denominator):

Solve 1/x = 4.

Excluded value: x = 0.

Multiply by x: 1 = 4x → x = 1/4.

Check: x = 1/4 ≠ 0. 1/(1/4) = 4. ✓

The big one — extraneous solution:

Solve x/(x − 2) = 2/(x − 2).

Excluded value: x = 2 (makes denominator zero).

LCD: (x − 2). Multiply both sides:
x = 2.

Check: x = 2 is the excluded value — plug it into the original and the denominator is zero. Undefined.

No solution. The only candidate was extraneous.

Memory hook: Multiplying by the LCD can introduce solutions that make the original denominator zero. An extraneous solution checks out algebraically but fails the domain test. Always check — always.

Segment 6 — More Solving + Work/Rate Applications (18 min)

Solving rational equation (work-rate setup):

Pipe A fills a tank in 3 hours; Pipe B fills the same tank in 6 hours. Working together, how long does it take?

Rate model: A contributes 1/3 of the tank per hour; B contributes 1/6.
Together: 1/3 + 1/6 = 1/t (where t is hours together).

LCD = 6t. Multiply through:
2t + t = 6
3t = 6
t = 2 hours.

Check: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2 of the tank per hour → 2 hours to fill it. ✓

Why work/rate problems are rational equations (make the connection explicit):

Any "person/machine completes a job at rate 1/T" setup produces fractions with variables in denominators — a rational equation. The LCD-clearing method solves them all.

Named misconception — "add the times, not the rates":

❌ Students add 3 + 6 = 9 hours. Wrong. Times don't add; rates add (1/T₁ + 1/T₂ = 1/T_together).

Segment 7 — Technology Workflow + AI-Critique Moment (12 min) · Session 2 continues

Technology workflow — verify a simplification in Desmos (exact steps):
1. Open desmos.com/calculator.
2. Line 1: type the original rational expression, e.g., (x^2 - 9) / (x + 3).
3. Line 2: type the simplified form, e.g., x - 3.
4. The graphs match (same line with a hole at x = −3) → your simplification is correct.
- If they don't match, you made a simplification error — Desmos shows you the fact, not the fix.
- Same trick for checking an addition: graph the original sum and the combined expression — if they're the same curve, you're right.

AI-critique moment (signature — students judge, not consume):

Paste this to an approved chatbot: "Simplify (x² + 5x + 6) / (x + 2) and show every step."

Then check its work. The correct answer: x² + 5x + 6 = (x + 2)(x + 3), so the result is (x + 3), x ≠ −2.

Common chatbot failure: the model may write (x² + 5x + 6) / (x + 2) = x + 5 + 6/(x+2) (using polynomial long division rather than factoring and canceling) — or it may try to cancel terms rather than factors and arrive at a wrong answer. If the model's answer isn't (x + 3), it's wrong — and you can explain exactly why.

The habit all semester: the tool drafts; you judge. Rational expressions are especially prone to chatbot term-cancellation errors.

Segment 8 — Callback, Hand-off & Next-Week Tease (7 min) · Session 2 closes (~77 min)

Callback to prior weeks:
- "Week 6 gave you the factoring toolkit — difference of squares, trinomials, GCF. Without that, rational expressions are impossible. This week you used every one of those tools."
- "Week 10 introduced rational functions and their graphs. The domain restrictions we found there — the vertical asymptotes — are exactly the excluded values we've been checking for extraneous solutions."

Next-week tease: "Week 12 switches from rational to radical expressions. Same idea, same danger: a different kind of 'clearing' operation (squaring both sides) that also produces extraneous solutions. The check-your-answers discipline you built this week carries straight into it."

Hand-off — this week's graded work:
- Lecture Tutorial 11 (AI tutor, share-link submission).
- Quiz 11 (end of week, no AI).
- Discussion 11 — "The Illegal Cancellation" (initial post by Fri Nov 13).
- Assignment 11 — "Fractions All the Way Down" (AI-coached, self-scored).

Instructor FAQ — Common Stumbles

Student does / says	Quick cure
Cancels (x + 3)/x → 3.	"The x is a term, not a factor. Factor the numerator first — if x divides every term as a common factor, pull it out. Then you can cancel. (x + 3) has no common factor of x."
Writes (x² − 9)/(x + 3) = x + 3 (flips the sign).	"Difference of squares: x² − 9 = (x+3)(x−3). Cancel (x+3) and you get x − 3, not x + 3."
LCD = product of denominators (doesn't factor).	"Factor the denominators first. The LCD uses each distinct factor to its highest power. Factoring shows you what's shared, which keeps the LCD as small as possible."
Forgets to check for extraneous solutions.	"Before you solve, write down the excluded values. After you solve, hold each answer up against that list. One check, every time — takes ten seconds and saves your grade."
Adds rates as times (work-rate).	"1/3 + 1/6 = 1/t — rates add, not times. Set up the fractions first, then solve."
Sign error when subtracting (A − B where B has multiple terms).	"Subtract the whole numerator of B — distribute the negative to every term: A − (b₁ + b₂) = A − b₁ − b₂. Write the parentheses explicitly if you keep dropping a sign."

Scope flag

This outline stays within Objective 7 (rational expressions and equations) and covers the topics needed for the week's assessments. Complex rational expressions (fractions within fractions) are mentioned briefly in the readings but are not assessed this week. The work-rate application connects to a standard Objective 7 context and is included in Assignment 11.

~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com