BetterGrades Precalculus · Unit 1 · Lesson

Rational expressions and restrictions repair

Simplify and solve rational expressions while preserving every restriction from the original form.

Opening

Start with the situation

A rational expression is a quotient, so every original denominator zero is excluded even when a common factor cancels.

These algebra choices are the load-bearing steps beneath later work with functions. A restriction lost here can turn into a false intercept, a missing asymptote, or an invalid model several lessons later.

Before you begin

Prerequisite check

Use signed arithmetic accurately.
Show algebraic steps in a checkable order.
State restrictions before simplifying.

Core explanation

Explanation

Record restrictions, factor completely, cancel only common factors, combine over a least common denominator, and check equation candidates against the original domain.

A divisor rational expression must be nonzero. Cancellation is division by a common factor, not removal of matching terms across addition.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through legal versus illegal cancellation, original-domain overlay, or another equivalent representation.

Conceptual reading

What the idea is really doing

Precalculus is unforgiving about hidden algebra errors. The useful habit is to separate reversible algebra from steps that can create candidates, and to keep domain restrictions beside the work instead of trying to remember them at the end.

This lesson narrows that lens to one goal: simplify and solve rational expressions while preserving every restriction from the original form. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Record restrictions.
Factor completely.
Cancel only common factors.
Combine over a least common denominator.

Verification: Re-read the original statement, not only the simplified line. Confirm every restriction, substitute each candidate, and describe what the result means before moving on.

Foundation walkthrough

Plan before calculating

Problem

Simplify $\frac{x^2+x-6}{x^2-4}$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Record restrictions, factor completely, cancel only common factors, combine over a least common denominator, and check equation candidates against the original domain.
Conclusion: The result is $\frac{x+3}{x+2},$ with $x\ne 2,-2$ .
Why the check works: The cancelled factor at $x=2$ remains an exclusion.

Worked examples

See the idea in three forms

foundation example

Simplify $\frac{x^2+x-6}{x^2-4}$

SolutionThe result is $\frac{x+3}{x+2},$ with $x\ne 2,-2$ .

The cancelled factor at $x=2$ remains an exclusion.

representation example

Simplify $\frac{x^2-16}{x-4}$

Solution $x+4, x\ne 4$

This example expresses rational expressions and restrictions repair in a second form.

transfer example

Solve $\frac{1}{x}=\frac{1}{5}$

Solution $x=5$

A divisor rational expression must be nonzero. Cancellation is division by a common factor, not removal of matching terms across addition.

Legal versus illegal cancellation. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The cancelled factor at x=2 remains an exclusion. — **Anchor figure · Legal versus illegal cancellation**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The cancelled factor at $x=2$ remains an exclusion.

Original-domain overlay. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational expressions and restrictions repair. — **Mechanism figure · Original-domain overlay**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational expressions and restrictions repair.

Candidate filter. Compare the valid path with the tempting shortcut. The figure shows why erasing the original restriction after cancellation leads to a false conclusion. — **Comparison and error figure · Candidate filter**
Compare the valid path with the tempting shortcut. The figure shows why erasing the original restriction after cancellation leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is erasing the original restriction after cancellation.

Check yourself

State restrictions of $\frac{5}{x(x-3)}$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

State restrictions of $\frac{5}{x(x-3)}$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Simplify $\frac{x^2-16}{x-4}$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Solve $\frac{1}{x}=\frac{1}{5}$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Why can $\frac{x+4}{x}$ not become $4$ ?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: The result is $\frac{x+3}{x+2},$ with $x\ne 2,-2$ . Use the foundation problem as evidence: Simplify $\frac{x^2+x-6}{x^2-4}$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of rational expressions and restrictions repair that it illustrates: Simplify $\frac{x^2-16}{x-4}$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is erasing the original restriction after cancellation.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Simplify $\frac{x^2+x-6}{x^2-4}$ . Describe what a graph, table, mapping, or algebraic form would have to show.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: Solve $\frac{1}{x}=\frac{1}{5}$ . Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for rational expressions and restrictions repair, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Radicals and rational exponents repair, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Stitz and Zeager, Precalculus Chapter 0
BetterGrades Algebra course
Redden, Advanced Algebra

No long source passage is reproduced.

Simplifyx2+x−6x2−4\frac{x^2+x-6}{x^2-4}x2−4x2+x−6​

Simplifyx2+x−6x2−4\frac{x^2+x-6}{x^2-4}x2−4x2+x−6​

Simplifyx2−16x−4\frac{x^2-16}{x-4}x−4x2−16​

Solve1x=15\frac{1}{x}=\frac{1}{5}x1​=51​

Original BetterGrades manuscript, rights-separated references.

Simplify $\frac{x^2+x-6}{x^2-4}$

Simplify $\frac{x^2+x-6}{x^2-4}$

Simplify $\frac{x^2-16}{x-4}$

Solve $\frac{1}{x}=\frac{1}{5}$