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Solving Linear Inequalities
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle).
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Solving Linear Inequalities
What is it?
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle).
In simple words
- Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule.
- THE ALGEBRA: 6x-7 >= 2x+17 -> 4x >= 24 -> x >= 6.
- THE NUMBER LINE: x<3: open circle at 3, ray left.
- Worked-method reminder: number line: filled circle at 6, ray to the right
Important terms
SOLVING LINEAR INEQUALITIES: Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle).
THE ALGEBRA: 6x-7 >= 2x+17 -> 4x >= 24 -> x >= 6. 9x+6 < 8x-2 -> x < -8. Fractions: multiply through by a positive LCD so you do not flip by accident.
THE NUMBER LINE: x<3: open circle at 3, ray left. x>=-2: filled circle at -2, ray right. The circle is the endpoint story; the ray is the solution set.
What you should learn
- Explain solving linear inequalities using the correct definitions, conditions, and notation.
- Apply collect like an equation, open vs filled correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Examples
- x>=6
- x<-8
- LCD positive: no flip
THE ALGEBRA
6x-7 >= 2x+17 -> 4x >= 24 -> x >= 6. 9x+6 < 8x-2 -> x < -8. Fractions: multiply through by a positive LCD so you do not flip by accident.
THE NUMBER LINE
x<3: open circle at 3, ray left. x>=-2: filled circle at -2, ray right. The circle is the endpoint story; the ray is the solution set.
Recommended for Class 10
Full school-level explanation
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle).
Solve, then draw:
WHAT — SOLVING LINEAR INEQUALITIES
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle). The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Solving Linear Inequalities matters because inequalities describe ranges and limits rather than one exact value. They are used for constraints, graphs, optimisation, and real situations such as budgets, dimensions, and capacity.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through THE ALGEBRA, then THE NUMBER LINE. Worked: solve 6x - 7 >= 2x + 17. 6x - 7 >= 2x + 17 6x - 2x >= 17 + 7 4x >= 24 x >= 6 number line: filled circle at 6, ray to the right Finish by checking signs, restrictions, units, and whether the answer matches the question.
THE ALGEBRA
6x-7 >= 2x+17 -> 4x >= 24 -> x >= 6. 9x+6 < 8x-2 -> x < -8. Fractions: multiply through by a positive LCD so you do not flip by accident.
Remember
- same steps as equations
Examples
- x>=6
- x<-8
- LCD positive: no flip
THE NUMBER LINE
x<3: open circle at 3, ray left. x>=-2: filled circle at -2, ray right. The circle is the endpoint story; the ray is the solution set.
Remember
- open = not equal
Examples
- x<3 open left
- x>=-2 filled right
- Not two open ends unless a compound
Remember
- collect like an equation
- open vs filled
- x >= 6
- all reals or empty
- same steps as equations
- open = not equal
- collect like an equation
- open vs filled
How the ideas connect
Worked: solve 6x - 7 >= 2x + 17. 6x - 7 >= 2x + 17 6x - 2x >= 17 + 7 4x >= 24 x >= 6 number line: filled circle at 6, ray to the right
Deeper understanding
Go beyond memorising Solving Linear Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: solve 6x - 7 >= 2x + 17. 6x - 7 >= 2x + 17 6x - 2x >= 17 + 7 4x >= 24 x >= 6 number line: filled circle at 6, ray to the right
Common mistakes
Exam focus
Core Idea
THE ALGEBRA
THE NUMBER LINE
Worked Method
Exam focus
Quick revision
- collect like an equation
- open vs filled
- x >= 6
- all reals or empty
- SOLVING LINEAR INEQUALITIES: Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule.
- THE ALGEBRA: 6x-7 >= 2x+17 -> 4x >= 24 -> x >= 6. 9x+6 < 8x-2 -> x < -8. Fractions: multiply through by a positiv
If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
For long answers on Solving Linear Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- collect like an equation
- open vs filled
- x >= 6
- all reals or empty
Important formulas
- collect like an equation
- open vs filled
- x >= 6
Common exam traps
- If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
- Not two open ends unless a compound
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
- Mistake
Not two open ends unless a compound
- Mistake
Do not treat THE ALGEBRA and THE NUMBER LINE as the same step; each has a different purpose.
- Mistake
If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
collect like an equation
open vs filled
x >= 6
all reals or empty
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive in
x >= 6
- Definition
- Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule. The solution is usually a ray: x < k or x >= k. Inclusive in
- Formula
- x >= 6
- Example
- x>=6
- Critical warning
- If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
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Exam Practice
Define
Q1. Define Solving Linear Inequalities.
Explain
Q2. Explain: THE ALGEBRA.
Example
Q3. Give one clear example of Solving Linear Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Solving Linear Inequalities.
Exam note
Q5. Write a short exam note on Solving Linear Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.