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Linear Inequalities
A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0. The solution set is all replacements of x that make the statement true, not a single root. Strict inequalities use < or > and do not include the boundary. Closed inequalities use <= or >= and do include the boundary. Equivalent inequalities have the same solution set.
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Linear Inequalities
What is it?
A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0. The solution set is all replacements of x that make the statement true, not a single root. Strict inequalities use < or > and do not include the boundary. Closed inequalities use <= or >= and do include the boundary. Equivalent inequalities have the same solution set.
In simple words
- A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0.
- A strict inequality uses < or >.
- A closed inequality uses <= or >=.
- The solution set is the collection of all replacements that work.
Important terms
linear inequality: A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0.
solution set: The solution set is all replacements of x that make the statement true, not a single root.
Strict: Strict inequalities use < or > and do not include the boundary.
Closed: Closed inequalities use <= or >= and do include the boundary.
What you should learn
- Explain linear inequalities in clear mathematical language
- Apply the ideas in strict < and >
- Solve problems involving closed <= and >=
Examples
- x < 4 (4 not included)
- 2x > 10 so x > 5
- x-3 < 0 so x < 3
STRICT < AND >
A strict inequality uses < or >. The boundary value that would make both sides equal is not in the solution set. For 3x-5 < 7 you get 3x < 12 so x < 4: 4 itself fails because 3(4)-5=7, which is not less than 7. On a number line, draw an open circle at the boundary.
CLOSED <= AND >=
A closed inequality uses <= or >=. The boundary that makes both sides equal is included. For 3x-5 <= 7 you still compute x=4 as the edge, but now 4 is allowed. On a number line, draw a filled circle at the boundary. Closed and strict can share the same working until the last symbol.
Recommended for Class 9
Full school-level explanation
A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0. The solution set is all replacements of x that make the statement true, not a single root. Strict inequalities use < or > and do not include the boundary. Closed inequalities use <= or >= and do include the boundary. Equivalent inequalities have the same solution set.
Read the symbol before you shade a number line:
What it means
A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0.
Main ideas
1. STRICT < AND >: A strict inequality uses < or >. 2. CLOSED <= AND >=: A closed inequality uses <= or >=.
Why it matters
The solution set is the collection of all replacements that work. For x < 4 that set is every real number to the left of 4, not a list of integers. Equivalent inequalities are two statements with that same set, obtained by legal moves.
How to answer in an exam
Do not treat an inequality like an equation and write a single answer x=4. Report the full solution set, and say whether the edge is in or out. Strict < > leave the edge out; closed <= >= keep it.
STRICT < AND >
A strict inequality uses < or >. The boundary value that would make both sides equal is not in the solution set. For 3x-5 < 7 you get 3x < 12 so x < 4: 4 itself fails because 3(4)-5=7, which is not less than 7. On a number line, draw an open circle at the boundary.
Remember
- strict inequality
- < or >
- not in the solution set
Examples
- x < 4 (4 not included)
- 2x > 10 so x > 5
- x-3 < 0 so x < 3
CLOSED <= AND >=
A closed inequality uses <= or >=. The boundary that makes both sides equal is included. For 3x-5 <= 7 you still compute x=4 as the edge, but now 4 is allowed. On a number line, draw a filled circle at the boundary. Closed and strict can share the same working until the last symbol.
Remember
- closed inequality
- <= or >=
- included
Examples
- x <= 4 (4 included)
- 2x >= 10 so x >= 5
- x-3 >= 0 so x >= 3
Remember
- linear inequality
- solution set
- Strict
- Closed
- Equivalent inequalities
- solution set
- all replacements that work
- Equivalent inequalities
How the ideas connect
The solution set is the collection of all replacements that work. For x < 4 that set is every real number to the left of 4, not a list of integers. Equivalent inequalities are two statements with that same set, obtained by legal moves.
Deeper understanding
Go beyond memorising Linear Inequalities. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
The solution set is the collection of all replacements that work. For x < 4 that set is every real number to the left of 4, not a list of integers. Equivalent inequalities are two statements with that same set, obtained by legal moves.
Common mistakes
Exam focus
Core Idea
Key Terms
Main Pattern
Worked Reminder
Exam focus
Quick revision
- linear inequality
- solution set
- Strict
- Closed
- linear inequality: A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0
- solution set: The solution set is all replacements of x that make the statement true, not a single root.
Do not treat an inequality like an equation and write a single answer x=4. Report the full solution set, and say whether the edge is in or out. Strict < > leave the edge out; closed <= >= keep it.
For long answers on Linear Inequalities: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- linear inequality
- solution set
- Strict
- Closed
Important formulas
- linear inequality
- solution set
- Strict
Common exam traps
- Do not treat an inequality like an equation and write a single answer x=4.
- Report the full solution set, and say whether the edge is in or out.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Do not treat an inequality like an equation and write a single answer x=4.
- Mistake
Report the full solution set, and say whether the edge is in or out.
- Mistake
Do not mix up strict < and > with closed <= and >=; use the condition given in the question.
- Mistake
Do not treat an inequality like an equation and write a single answer x=4. Report the full solution set, and say whether the edge is in or out. Strict < > leave
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
linear inequality
solution set
Strict
Closed
A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0. The solution set is all replacements of x that make the st
<= or >=
- Definition
- A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0. The solution set is all replacements of x that make the st
- Formula
- <= or >=
- Example
- x < 4 (4 not included)
- Critical warning
- Do not treat an inequality like an equation and write a single answer x=4.
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Exam Practice
Define
Q1. Define Linear Inequalities.
Explain
Q2. Explain: STRICT < AND >.
Example
Q3. Give one clear example of Linear Inequalities and say why it fits.
Mistake
Q4. State one common mistake students make about Linear Inequalities.
Exam note
Q5. Write a short exam note on Linear Inequalities: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.