BetterGrades Precalculus · Unit 6 · Lesson
Rational equations and function intersections
Solve rational equations and intersections exactly or numerically while rejecting excluded candidates.
Start with the situation
Rational intersections satisfy 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.
Prerequisite check
- Factor numerator and denominator.
- Preserve original restrictions.
- Use sign and asymptotic notation.
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.
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.
Plan before calculating
Problem
Solve
- 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 is excluded, so no solution.
- Why the check works
- The transformed equation exists where the original does not.
See the idea in three forms
foundation example
Solve
SolutionCandidate is excluded, so no solution.
The transformed equation exists where the original does not.
representation example
Solve
Solution
This example expresses rational equations and function intersections in a second form.
transfer example
X-intercept of .
Solution
An excluded candidate can arise because the clearing multiplier is zero where the original equation is undefined.
Read this graph as text
Rational equations and function intersections · 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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve rational equations and intersections exactly or numerically while rejecting excluded candidates.
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.
Read this graph as text
Rational equations and function intersections · 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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve rational equations and intersections exactly or numerically while rejecting excluded candidates.
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for rational equations and function intersections.
Read this graph as text
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. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Solve rational equations and intersections exactly or numerically while rejecting excluded candidates.
Compare the valid path with the tempting shortcut. The figure shows why accepting every solution of the cleared equation leads to a false conclusion.
Find the first invalid move
A frequent error is accepting every solution of the cleared equation.
Solve
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10 concrete questions
01Solve
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02Solve
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03X-intercept of .
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04How find intersection ?
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05Explain why this conclusion is valid: Candidate is excluded, so no solution. Use the foundation problem as evidence: Solve .
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06Solve the representation example, then name the feature of rational equations and function intersections that it illustrates: Solve
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07Correct this reasoning and identify the first unsafe assumption: A frequent error is accepting every solution of the cleared equation.
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08Connect two representations for this example: Solve . Describe what a graph, table, mapping, or algebraic form would have to show.
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09Create a nearby example by changing one number or condition in this prompt: X-intercept of . Predict the effect, solve your new example, and compare it with the original.
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10Write a short verification checklist for rational equations and function intersections, then apply it to one worked example from this lesson.
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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.