MathematicsChapter 5: Linear Equations and Inequalities
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.
Learning objectives
- Explain linear inequalities in clear mathematical language
- Apply the ideas in strict < and >
- Solve problems involving closed <= and >=
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 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.
Where / symbols
- 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.
Overview
Read the symbol before you shade a number line:
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.
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.
Examples x <= 4 (4 included) · 2x >= 10 so x >= 5 · x-3 >= 0 so x >= 3
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.
Explanation
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.
In simple words
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.
Key points
- linear inequality
- solution set
- Strict
- Closed
- strict inequality
- < or >
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Linear Inequalities
Sticky Notes · Quick Revision
Core Idea
- A linear inequality in one variable looks like ax+b < 0, ax+b > 0, ax+b <= 0, or ax+b >= 0 with a!=0.
Key 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.
Main Pattern
- 1. STRICT < AND >: A strict inequality uses < or >.
- 2. CLOSED <= AND >=: A closed inequality uses <= or >=.
Worked Reminder
- x < 4 (4 not included)
- 2x > 10 so x > 5
- x-3 < 0 so x < 3
Common 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 the edge out; closed <= >= keep it.
Exam Tip
- State the rule for linear inequalities, show each algebraic step, and check the final answer against the question.
60-second revision
- 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.
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