# Book Isle absolute-value equations: recover both points three units from two
Language: en
Canonical: [Book Isle absolute-value equations: recover both points three units from two]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-13T00:00:00+08:00
Modified: 2026-11-13T09:43:51+08:00
Author: Bison
Use a number line, branches and substitution, checking the right-hand sign before counting solutions.
English translation of the [Chinese original](), translated on 2026-11-13. Original publication dates, attribution, code and version notes are retained.
Do not erase absolute-value bars and keep only one branch. Read an original card in Book Isle, find both points three units from two, then test zero and negative right sides. Calculate on paper or in available notes; the app supplies reading/retrieval, not an assumed equation solver or automatic number line.
## 1. Preserve the full question and real-number domain
Save an original card in an already readable format, explicitly seeking real-number solutions without hiding answers in its filename. Check paired bars, minus and equals signs after opening; record book/chapter/anchor. Use external notes if an expected note control is absent. Preserve originals and read independently before answers, avoiding bars misread as digits or omission of the center two, which would create a different equation.
## 2. Transcribe three right-hand cases
The equations differ only on the right. Copy each separately and note that distance cannot be negative. Do not silently change minus three into plus three or share one solution set across similar lines. The bars enclose x minus two, locating the center, rather than x alone. Leave answers blank and return to the displayed question. Repair an unreadable copy before calculating instead of replacing the question from memory.
```
A: |x - 2| = 3
B: |x - 2| = 0
C: |x - 2| = -3
Domain: real numbers
```
## 3. Mark equal distances on both sides of the center
Use the distance interpretation in [OpenStax’s absolute-value functions and equations](). Mark center two, then move three units right to five and left to minus one. Equal distances do not make the points equal or require nonnegative x. Label scale and count intervals, not tick marks. This is a hand-drawn diagram, not an automatic Book Isle feature. Bind it to the question anchor so the reference is clearly two rather than zero. Put center coordinate two and distance three in separate fields; together they are not the decimal 2.3. Both sides must use the same scale rather than widening only the right-hand intervals while calling both distances three units.
## 4. Solve both branches and substitute each
A gives x−2=3 or x−2=−3, yielding five and minus one. In the original equation, their inner values are three and minus three, both with absolute value three. Rejecting the negative inner value confuses values before and after absolute value. Substitute both candidates and list two distinct real solutions; do not chain alternative branches as if they were simultaneously equal.
## 5. Count zero and negative-distance cases separately
B has zero distance, so only center two qualifies. Both branch writings yield the same value, counted once. C has a negative right side while its real absolute-value left side is nonnegative, so there is no solution—not x=−3 or a surviving positive solution. Check the right side before branching. Explain that zero/one/two counts distinct real solutions, not written lines.
## 6. Change the grouping to test question identity
For a grouping control, use absolute value of x, then subtract two, equal to three. Rearrangement gives |x|=5, with five and minus five. Different bar scope changes the result, so A’s minus one does not transfer. Describe the two scopes and test a negative candidate in each, distinguishing grouping from arithmetic errors. Preserve the original; consult a clear edition rather than inferring an author’s bar placement from the expected answer.
## 7. Troubleshoot missing branches and rejected negatives
Five alone suggests a missing left-side branch. Deleting minus one suggests adding an unstated x≥0 restriction. Two roots for C suggest overlooking nonnegative distance. Substitute into the original, not merely an incorrectly derived branch. Change one error and preserve its reason. Record only real retrieval/note actions in Book Isle; do not invent automated feedback or delete old notes to make prior reasoning appear correct.
## 8. Return and recheck with answers covered
Leave the reading location and return by its recorded anchor, confirming the same card and grouping. Cover answers and redraw center two with equal three-unit distances, explaining two/one/zero distinct real solutions for A/B/C. Compare notes and substitute every candidate into its original equation. Keep question, diagram, branches, checks and corrections, with unresolved rendering issues separate. Continue preserving the original book, original dates and original excerpts. Reproducible reasoning is not a claim that Book Isle proves all absolute-value equations.
## Related tutorials and sources
[Radicals and absolute value](), [Worked-example symbol records]().
Checked on November 13, 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.