Book Isle rational equations: cancellation does not restore excluded values

A canceled factor does not make an originally undefined value valid. Prepare an original short reading page, use Book Isle for reading, and retain domain, transformations and original-equation checks in available notes or externally. Readers perform the algebra; no automatic solving or grading control is assumed.

1. Record the original equation and denominator

Write (x²−1)/(x−1)=2 over the reals. Identify x−1 and record x≠1 before cancellation. Keep parentheses clear and never erase the original restriction after obtaining a polynomial expression. Use an available authorized reading format or external monospaced notes when formula handling is limited, without modifying an existing book.

2. Factor and cancel within the restriction

Factor the numerator as (x−1)(x+1) and cancel the nonzero common factor only for x≠1, obtaining x+1=2. OpenStax’s rational-equation section supports retaining denominator restrictions and checking the original. Repeat the restriction beside the transformation rather than treating cancellation as a definition at the excluded point. As a second route, clearing the denominator gives x²−1=2x−2, then (x−1)²=0. It produces the same excluded candidate. The issue is retaining premises, not a particular cancellation gesture. Keep this as a cross-check, not a second solution.

3. Separate a candidate from the solution set

The transformed equation gives candidate x=1, excluded by the original denominator. The original solution set is empty; 0/0 cannot be canceled into two. Keep candidate, original definedness and retention on separate lines. An empty set is a mathematical result, not missing app output.

Original: (x^2-1)/(x-1)=2, x!=1
Within domain: x+1=2 -> candidate x=1
Original denominator at 1: 0
Original solution set: empty

4. Change only the right side to three

Change only the right side to three. The restriction stays x≠1, and x+1=3 gives x=2. Substitution yields (4−1)/(2−1)=3, so the solution set is {2}. Label both cases separately: cancellation does not make every candidate invalid, and checking the original determines retention.

Control: (x^2-1)/(x-1)=3, x!=1
x+1=3 -> candidate x=2
Original at 2: (4-1)/(2-1)=3
Solution set: {2}

5. Do not fill an excluded point with nearby values

At x=0 the original expression is 1, and at x=2 it is 3. These checks confirm agreement within the domain without filling x=1. This task does not redefine the function or infer definedness from nearby continuity. If another text explicitly defines a new function at that point, name it separately rather than merging it with the original expression. Original-equation evidence remains decisive.

6. Keep a recoverable reading anchor in Book Isle

Open the owned page, identify right side two versus three and save domain/candidate/check separately in available notes or externally. Record chapter/page plus a short anchor rather than a resize-sensitive screen line. Reopen both cases. If math formatting is unavailable, use parenthesized plain text while retaining the source instead of treating missing superscripts as algebra errors.

7. Troubleshoot lost restrictions and invalid cancellation

If one is retained, check lost restrictions. If factorization gives x−1 alone, recheck the difference of squares. Cancel multiplicative factors, not separate terms. Diagnose parentheses, powers and signs before changing the conclusion; retain the original error and reason rather than rewriting the problem. These are reader checks, not automatic grading.

8. Reopen and deliver both complete cases

Reopen the page and recompute both with conclusions hidden. Deliver originals, real domain, excluded value, factors, candidates and substitution checks. Explain why the same structure yields an empty set versus {2}. Recompute restrictions for new equations and leave unperformed importing/device steps pending.

Related tutorials and sources

Radicals and original-equation checks, Absolute-value conditions.

Checked on November 14, 2026 against the official Book Isle App Store listing and Apple public lookup: Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. Use existing reading, available notes or external records and verify controls in your installed version. Original samples and manual calculations are not new automatic analysis features. No physical-device, private-file or universal export-environment testing is claimed.