# Read inequalities in Book Isle: check direction and endpoints when dividing by a negative

Language: en
Canonical: [Read inequalities in Book Isle: check direction and endpoints when dividing by a negative](<https://allluckly.cn/en/guides/nasbook-linear-inequality-negative-factor-direction/>)
中文：[中文 HTML](<https://allluckly.cn/guides/nasbook-linear-inequality-negative-factor-direction/>)
English: [English HTML](<https://allluckly.cn/en/guides/nasbook-linear-inequality-negative-factor-direction/>)
Published: 2026-11-12T00:00:00+08:00
Modified: 2026-11-12T09:43:29+08:00
Author: Bison

Transform an original inequality step by step, checking its boundary and values on both sides, and distinguish an unknown coefficient whose sign prevents direct division.

English translation of the [Chinese original](<https://allluckly.cn/guides/nasbook-linear-inequality-negative-factor-direction/>), translated on 2026-11-12. Original publication dates, attribution, code and version notes are retained.

An inequality reverses when multiplying or dividing by a negative, not merely when moving to a new line or seeing a minus sign. Follow a linear inequality in Book Isle and use substitutions to compare the original and transformed statements. Record the algebra and number line manually; the app need not solve it, and one satisfying value does not describe the whole solution set.

## Keep the complete statement and domain

Record the book, edition, section and example number. Here x is real and the original is `-2*x+1>7`. An asterisk is simply an external plain-text multiplication sign, not a formula-recognition feature. The greater-than relation is strict. Copy negative two, plus one and seven, reading multiplication before addition, rather than multiplying the entire x-plus-one quantity. An integer-only problem would require a later restriction; do not add it here. Keep the source statement separate from your shorthand.

## Add or subtract equally without reversing

Subtract one from both sides to obtain `-2*x>6`; the relation remains greater than. Annotate subtract one on both sides instead of merely saying move the one, so the operation is visible. Do not reverse here just because the coefficient is already negative: this step has not multiplied or divided both sides by a negative. Computing seven minus one as eight is an arithmetic error; subtracting only on the left is a different error. Identify the actual step rather than grouping both as moving terms incorrectly.

## Reverse when dividing by the known negative

Divide both sides by negative two. Six divided by negative two is negative three, giving `x<-3`. Multiplying two compared numbers by negative one reflects their positions on the number line and reverses their order; division by negative two includes that negative scaling. Do not retain greater-than or change only the number’s sign. The divisor is known and nonzero here. Division by zero would be undefined regardless of whether the relation was reversed; changing the symbol cannot repair that operation.

## Check the boundary and a value on each side

At x equal to negative three, the original left side is seven, so seven greater than seven is false and the boundary is excluded. At negative four it is nine, which satisfies the inequality; at negative two it is five, which does not. These check the boundary and values inside and outside the candidate set, not all real numbers exhaustively. Also substitute non-integer negative 3.5: the original left side is eight, greater than seven, so it belongs to the real solution set. Do not list only integers such as negative four and negative five. Negative two immediately refutes an incorrect greater-than-negative-three answer. Show multiplication before addition: negative two times negative four is positive eight. Losing that sign is not a failure of the inequality rule.

## Show strictness with an open boundary

On your number line, mark negative three as an open endpoint and shade toward smaller values, using interval `(-∞,-3)`. Negative infinity is not an attainable real endpoint. In a separate control, change only the original relation to greater than or equal to seven; the result is `x<=-3`, including negative three. Keep the two endpoint conventions distinct instead of copying one line. When enlarging the text, check whether the small equality stroke exists. If the source remains unclear, mark it unresolved rather than inventing a non-strict relation from a familiar answer.

## Split cases for an unknown coefficient

For a control `c*x>6` with c’s sign unspecified, do not simply write x greater than six divided by c. Positive c preserves direction; negative c reverses it. If c is zero, the left side is zero and zero greater than six is false, so there are no real solutions. State all three conditions. Nonzero does not mean positive. The original permits direct division because negative two has a known sign. If the book assumes positive c, retain that additional premise and do not carry it automatically into another problem.

## Locate the first differing transformation

When answers differ, compare the original, subtract-one line, divide-by-negative-two line and endpoint line. Do not reason backwards from an attractive final answer. A correct negative-three boundary with wrong direction points to division; a filled endpoint points to strictness; listing only negative-four, negative-five and other integers points to the domain. Keep the incorrect version as a personal control without altering the book or reporting your algebra mistake as reader incompatibility. Check problem numbers in search results: a similar-looking inequality may have different conditions.

## Leave and return to repeat all three checks

Leave reading and return to the original. Without consulting notes, subtract one, divide by negative two, and substitute negative four, negative three and negative two again. Explain why subtraction preserves direction, negative division reverses it, strictness excludes the boundary, and unknown c needs cases. Retain the source location, real domain, operations, three results and endpoint. Success means every symbol has an operation or premise behind it, not an automatic app answer. A non-strict or integer-only variant needs its own checks rather than the old solution under a new title.

## Related guides and sources

[Book Isle: Align interval brackets, inequalities and number-line endpoints](<https://allluckly.cn/en/guides/nasbook-interval-endpoint-number-line/>); [Book Isle definition reading: necessary conditions, sufficient conditions and counterexamples](<https://allluckly.cn/en/guides/nasbook-definition-necessary-sufficient-boundary/>).

Checked on November 12, 2026: [Apple public information](<https://itunes.apple.com/lookup?id=6766198898&country=cn&entity=software&timestamp=20261112>) lists Book Isle 1.1.4, requiring iOS / iPadOS 15.6 or later. See the [official App Store](<https://apps.apple.com/cn/app/id6766198898>) for published capabilities; verify specific controls in the installed version.

Method reference: [OpenStax: solving linear inequalities](<https://openstax.org/books/elementary-algebra-2e/pages/2-7-solve-linear-inequalities>). Samples, passages and controls are original exercises. Sources establish definitions and checking methods rather than new automatic analysis in the app. Manual records may be kept externally, and unknown fields should remain explicit. This article does not claim step-by-step testing on a physical device, private server or every output environment.