BetterGrades Precalculus · Unit 6 · Lesson

Rational equations and function intersections

Solve rational equations and intersections exactly or numerically while rejecting excluded candidates.

Opening

Start with the situation

Rational intersections satisfy $f(x)=g(x),$ but denominator clearing creates candidates that must be checked in the original domains.

Rational graphs are organized around the inputs the denominator forbids. Those exclusions divide the graph into continuity intervals and explain why a simplified expression may still contain a hole or asymptote.

Before you begin

Prerequisite check

Factor numerator and denominator.
Preserve original restrictions.
Use sign and asymptotic notation.

Core explanation

Explanation

Record restrictions, multiply every term by an LCD, solve the resulting equation, and reject excluded candidates.

An excluded candidate can arise because the clearing multiplier is zero where the original equation is undefined.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through intersection equation, candidate filter, or another equivalent representation.

Conceptual reading

What the idea is really doing

A rational function carries permanent memory of its original denominator. Factoring may reveal holes, asymptotes, and sign changes, but cancellation never restores an input that the original formula excluded.

This lesson narrows that lens to one goal: solve rational equations and intersections exactly or numerically while rejecting excluded candidates. 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.
Multiply every term by an LCD.
Solve the resulting equation.
Reject excluded candidates.

Verification: Record exclusions first, then compare the factored and simplified forms. Test one point in every sign interval and examine both sides of each vertical asymptote.

Foundation walkthrough

Plan before calculating

Problem

Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Record restrictions, multiply every term by an LCD, solve the resulting equation, and reject excluded candidates.
Conclusion: Candidate $x=2$ is excluded, so no solution.
Why the check works: The transformed equation exists where the original does not.

Worked examples

See the idea in three forms

foundation example

Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$

SolutionCandidate $x=2$ is excluded, so no solution.

The transformed equation exists where the original does not.

representation example

Solve $\frac{2}{x}=\frac{1}{x-3}$

Solution $x=6$

This example expresses rational equations and function intersections in a second form.

transfer example

X-intercept of $\frac{x-2}{x+3}$ .

Solution $x=2$

An excluded candidate can arise because the clearing multiplier is zero where the original equation is undefined.

Intersection equation. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The transformed equation exists where the original does not. — **Anchor figure · Intersection equation**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The transformed equation exists where the original does not.

Candidate filter. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational equations and function intersections. — **Mechanism figure · Candidate filter**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational equations and function intersections.

Exact-numerical method tree. Compare the valid path with the tempting shortcut. The figure shows why accepting every solution of the cleared equation leads to a false conclusion. — **Comparison and error figure · Exact-numerical method tree**
Compare the valid path with the tempting shortcut. The figure shows why accepting every solution of the cleared equation leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is accepting every solution of the cleared equation.

Check yourself

Solve $\frac{1}{x}=\frac{1}{4}$

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

Ten concrete questions

Practice 101

Solve $\frac{1}{x}=\frac{1}{4}$

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 202

Solve $\frac{2}{x}=\frac{1}{x-3}$

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 303

X-intercept of $\frac{x-2}{x+3}$ .

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 404

How find intersection $y$ ?

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: Candidate $x=2$ is excluded, so no solution. Use the foundation problem as evidence: Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$ .

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 equations and function intersections that it illustrates: Solve $\frac{2}{x}=\frac{1}{x-3}$

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 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is accepting every solution of the cleared equation.

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: Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$ . 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: X-intercept of $\frac{x-2}{x+3}$ . 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 equations and function intersections, 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, Rational inequalities, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Lippman and Rasmussen, Precalculus Volume 1
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

Solvex+1x−2=3x−2\frac{x+1}{x-2}=\frac{3}{x-2}x−2x+1​=x−23​

Solvex+1x−2=3x−2\frac{x+1}{x-2}=\frac{3}{x-2}x−2x+1​=x−23​

Solve2x=1x−3\frac{2}{x}=\frac{1}{x-3}x2​=x−31​

X-intercept of x−2x+3\frac{x-2}{x+3}x+3x−2​.

Original BetterGrades manuscript, rights-separated references.

Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$

Solve $\frac{x+1}{x-2}=\frac{3}{x-2}$

Solve $\frac{2}{x}=\frac{1}{x-3}$

X-intercept of $\frac{x-2}{x+3}$ .