MathematicsChapter 4: Linear and Quadratic Inequalities
Solving Linear Inequalities
Collect x terms on one side, numbers on the other, like an equation, but remember the reverse rule.
Learning objectives
- 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.
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 inequalities use <= or >= (filled circle on a number line). Strict use < or > (open circle).
Where / symbols
- 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.
Overview
Solve, then draw:
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.
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.
Examples x<3 open left · x>=-2 filled right · Not two open ends unless a compound
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.
Worked example
- 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
In simple words
If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
Key points
- collect like an equation
- open vs filled
- same steps as equations
- open = not equal
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Solving Linear Inequalities
Sticky Notes · Quick Revision
Core Idea
- 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.
Worked Method
- 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 Misconception
- If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
FBISE Exam Check
- If both sides simplify to a true number sentence (1<=4), every real x works. If it becomes 1>=4, no solution.
60-second revision
- 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
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