Welch Teaching & Coaching

Trigonometry • College Preparatory Mathematics

Unit 9: Rational Expressions

Simplify and operate on rational expressions, solve rational equations, simplify complex fractions, and graph rational functions.

U9

Unit Road Map

Lesson Map

Use this table as your route through the unit. Lesson numbers jump directly to the matching explanation.

Lesson	Topic	You should be able to…
9.1	Simplifying Rational Expressions	Factor completely and cancel common factors while preserving restrictions.
9.2	Multiplication & Division	Factor, multiply by reciprocals when dividing, and simplify.
9.3	Addition & Subtraction: Related Denominators	Use an LCD when one denominator is already a factor of another.
9.4	Addition & Subtraction: Unlike Denominators	Factor denominators and build the least common denominator.
9.5	Complex Fractions	Clear smaller denominators using the LCD.
9.6	Rational Equations	State restrictions, clear denominators, solve, and check.
9.7	Graphing Rational Functions	Identify domain restrictions, intercepts, asymptotes, and end behavior.

Navigation

How the Unit Is Built

Each resource below is part of the same learning cycle. Select a card to jump to it.

Lesson MapSee the sequence and jump directly to any lesson.Content SummaryRead the ideas, methods, examples, and checks.Video LibraryOpen a targeted Khan Academy or YouTube search for each lesson.VocabularyReview the language needed to read and explain the mathematics.Sample ProblemsPractice with expandable worked solutions.Equations & RulesUse a compact reference for formulas and decision rules.Review GuidePrepare with a skill checklist and fresh mixed practice.Performance TaskApply the unit in a modeling or measurement context.

Learn

Content Summary

Work through these lessons in order. Focus on the meaning of the mathematics, the decision that selects a method, and a check that confirms the result.

Lesson 9.1

Simplifying Rational Expressions

Factor completely and cancel common factors while preserving restrictions.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.2

Multiplication & Division

Factor, multiply by reciprocals when dividing, and simplify.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.3

Addition & Subtraction: Related Denominators

Use an LCD when one denominator is already a factor of another.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.4

Addition & Subtraction: Unlike Denominators

Factor denominators and build the least common denominator.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.5

Complex Fractions

Clear smaller denominators using the LCD.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.6

Rational Equations

State restrictions, clear denominators, solve, and check.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Lesson 9.7

Graphing Rational Functions

Identify domain restrictions, intercepts, asymptotes, and end behavior.

What to understand: Explain what the method is doing, not only which buttons or steps produce an answer.

Check: Ask whether the sign, size, units, quadrant, restrictions, or graph behavior make sense before accepting the result.

Watch & Reinforce

Video Library

These links open focused video searches by lesson topic. Use them as reinforcement rather than as a substitute for showing your own reasoning.

Lesson 9.1

Simplifying Rational Expressions

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.2

Multiplication & Division

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.3

Addition & Subtraction: Related Denominators

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.4

Addition & Subtraction: Unlike Denominators

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.5

Complex Fractions

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.6

Rational Equations

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Lesson 9.7

Graphing Rational Functions

Use a short lesson after reading the website explanation, then return to the practice.

Find a matching video →

Language

Vocabulary

These terms should appear naturally in explanations, diagrams, and written reasoning.

rational expression
Write a definition in your own words and add an example or sketch in your STRIDES notes.

restriction
Write a definition in your own words and add an example or sketch in your STRIDES notes.

least common denominator
Write a definition in your own words and add an example or sketch in your STRIDES notes.

complex fraction
Write a definition in your own words and add an example or sketch in your STRIDES notes.

rational equation
Write a definition in your own words and add an example or sketch in your STRIDES notes.

extraneous solution
Write a definition in your own words and add an example or sketch in your STRIDES notes.

vertical asymptote
Write a definition in your own words and add an example or sketch in your STRIDES notes.

horizontal asymptote
Write a definition in your own words and add an example or sketch in your STRIDES notes.

hole
Write a definition in your own words and add an example or sketch in your STRIDES notes.

removable discontinuity
Write a definition in your own words and add an example or sketch in your STRIDES notes.

Practice

Sample Problems

Try each problem before opening the solution. The examples are original practice aligned to the unit skills.

1

Simplify

Simplify (x²−9)/(x²−x−6).

Show solution

Factor: (x−3)(x+3)/[(x−3)(x+2)] = (x+3)/(x+2), with x≠3,−2.

2

Multiply

Simplify (x²−4)/(x²+3x+2) · (x+1)/(x−2).

Show solution

Factor and cancel to get (x+2)/(x+2)=1, with original restrictions x≠−1,−2,2.

3

Divide

Simplify (x/3) ÷ (2x/9).

Show solution

x/3 · 9/(2x)=3/2, x≠0.

4

Add

Simplify 2/x + 3/(x+1).

Show solution

[2(x+1)+3x]/[x(x+1)] = (5x+2)/[x(x+1)].

5

Equation

Solve 1/x + 1/(x+2)=3/4.

Show solution

Multiply by 4x(x+2): 4(x+2)+4x=3x(x+2). Solve 3x²−2x−8=0 → x=2 or −4/3; both meet restrictions.

6

Graph feature

For f(x)=1/(x−4), identify the vertical asymptote.

Show solution

x=4.

Reference

Equations & Rules

Know what each relationship means and when it applies; do not treat the box as a list of disconnected formulas.

Restriction

Denominators cannot equal zero.

Multiply

Factor first; cancel common factors; multiply remaining factors.

Divide

Multiply by the reciprocal of the divisor.

Add/Subtract

Use a least common denominator before combining numerators.

Solve Equations

State restrictions → multiply by LCD → solve → check.

Vertical Asymptote

Often occurs at a non-canceled denominator zero.

Hole

A canceled common factor creates a removable discontinuity at its zero.

Prepare

Review Guide

Skill Checklist

I can factor completely and cancel common factors while preserving restrictions.
I can factor, multiply by reciprocals when dividing, and simplify.
I can use an LCD when one denominator is already a factor of another.
I can factor denominators and build the least common denominator.
I can clear smaller denominators using the LCD.
I can state restrictions, clear denominators, solve, and check.
I can identify domain restrictions, intercepts, asymptotes, and end behavior.

How to review

Complete the unit Skills Check packet without notes.
Mark the exact skill behind each error.
Return to that lesson and complete a fresh example.
Explain the correction in words, then retry a similar problem.
Finish with mixed practice so you must choose the method yourself.

Apply

Performance Task

Rational Model Investigation

Use a rational relationship such as average rate, inverse variation, resistance, density, or cost per person. Build and interpret a model, including restrictions and graph behavior.

Evidence to include

A labeled mathematical model, organized calculations, appropriate units and precision, a written interpretation, and a check or discussion of limitations.