Linear inequalities
An inequality describes a range of values. Adding or subtracting the same amount preserves order. Multiplying or dividing by a negative reverses order: 2 < 5 becomes −2 > −5 when both sides are multiplied by −1.
What this guide covers
Linear inequalities in one variable and solutions on a number line. Two-variable shaded regions are outside this guide.
Syllabus reference: 2026 Mathematics 4052, N7, page 7. These are Mentora search groupings, not new official syllabus categories.
The content in this guide was also checked against 2027 G3 Mathematics K310, N7, page 7 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.
Worked example
Solve 7 − 3x < 16 and show the solution on a number line.
Subtract 7: −3x < 9. Divide by −3 and reverse the sign: x > −3. Use an open circle at −3 and shade to the right. Test x = 0: 7 < 16 is true; the endpoint x = −3 gives 16 < 16, which is false.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
Solve 4x + 1 ≤ 13.
Show worked answer
Subtract 1 and divide by positive 4: x ≤ 3. Use a filled endpoint at 3 and shade left.
Check 2
Solve −2x ≥ 8.
Show worked answer
Divide by −2, reversing the sign: x ≤ −4. Use a filled endpoint at −4 and shade left. At x = −5, 10 ≥ 8 is true.
Check 3
List the integers satisfying −1 < 2x + 1 ≤ 7.
Show worked answer
Subtract 1 throughout: −2 < 2x ≤ 6. Divide by 2: −1 < x ≤ 3. The integers are 0, 1, 2, 3; −1 is excluded.
Mistakes to watch for
- Reverse the sign for multiplication or division by a negative, not simply whenever a negative number appears.
- Open endpoints mean strict < or >; filled endpoints include equality.
- Do not list only integer solutions unless the question restricts the variable to integers.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
Useful foundations to check
These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.
Applications and next topics
Use one-variable inequalities for budgets or capacity limits. Solving an equation finds a boundary; the inequality also says which side of that boundary is allowed.
Choose a useful next resource
To understand a recurring error rather than relearn the whole skill: Check signs when rearranging.
Sources and review
Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-12; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.