MathematicsChapter 4: Linear and Quadratic Inequalities
Word Linear Inequalities
Translate English into an inequality, then solve.
Learning objectives
- Explain word linear inequalities using the correct definitions, conditions, and notation.
- Apply translate, then solve correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
Translate English into an inequality, then solve. 'Four more than three times x is less than 40' is 3x+4<40. Area and weight limits become <= or >. The algebra is Chapter 4; the sentence is the extra step. Units stay in the sentence, not mixed into x blindly.
Where / symbols
- WORD LINEAR INEQUALITIES
- Translate English into an inequality, then solve. 'Four more than three times x is less than 40' is 3x+4<40. Area and weight limits become <= or >. The algebra is Chapter 4; the sentence is the extra step. Units stay in the sentence, not mixed into x blindly.
- PHRASES
- 'Less than' is <. 'At most / no more than' is <=. 'At least' is >=. Twice the sum of x and 8 is 2(x+8), not 2x+8 unless the 'twice' hits only x.
- AREA / WEIGHT
- Rectangle height 9, width x+2, area>81: 9(x+2)>81. Triangle area 1/2 * 8 * (x+1) <= 44. Bag 26.5 plus w at most 60: 26.5+w<=60.
Overview
Words to symbols:
Phrases
'Less than' is <. 'At most / no more than' is <=. 'At least' is >=. Twice the sum of x and 8 is 2(x+8), not 2x+8 unless the 'twice' hits only x.
Examples 3x+4<40 · 2(x+8)<=-2 · x+2 is a width, not an area
Area / Weight
Rectangle height 9, width x+2, area>81: 9(x+2)>81. Triangle area 12 8 (x+1) <= 44. Bag 26.5 plus w at most 60: 26.5+w<=60.
Examples 9(x+2)>81 -> x>7 · 4(x+1)<=44 -> x<=10 · w<=33.5
WHAT — WORD LINEAR INEQUALITIES
Translate English into an inequality, then solve. 'Four more than three times x is less than 40' is 3x+4<40. Area and weight limits become <= or >. The algebra is Chapter 4; the sentence is the extra step. Units stay in the sentence, not mixed into x blindly. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Word 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 PHRASES, then AREA / WEIGHT. Worked: 9 ft by (x+2) ft rectangle, area > 81. 9(x + 2) > 81 x + 2 > 9 x > 7 Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- 9 ft by (x+2) ft rectangle, area > 81.
- 9(x + 2) > 81
- x + 2 > 9
- x > 7
In simple words
Area of a triangle is (12) base height. Forgetting 12 doubles the bound on x.
Key points
- translate, then solve
- at most = <=
- write the formula first
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Word Linear Inequalities
Sticky Notes · Quick Revision
Core Idea
- Translate English into an inequality, then solve. 'Four more than three times x is less than 40' is 3x+4<40. Area and weight limits become <= or >. The algebra is Chapter 4; the sentence is the extra step. Units stay in the sentence, not mixed into x blindly.
PHRASES
- 'Less than' is <. 'At most / no more than' is <=. 'At least' is >=. Twice the sum of x and 8 is 2(x+8), not 2x+8 unless the 'twice' hits only x.
AREA / WEIGHT
- Rectangle height 9, width x+2, area>81: 9(x+2)>81. Triangle area 12 8 (x+1) <= 44. Bag 26.5 plus w at most 60: 26.5+w<=60.
Worked Method
- Worked: 9 ft by (x+2) ft rectangle, area > 81.
- 9(x + 2) > 81
- x + 2 > 9
- x > 7
Common Misconception
- Area of a triangle is (12) base height. Forgetting 12 doubles the bound on x.
FBISE Exam Check
- Area of a triangle is (12) base height. Forgetting 12 doubles the bound on x.
60-second revision
- Translate English into an inequality, then solve.
- PHRASES: 'Less than' is <.
- AREA / WEIGHT: Rectangle height 9, width x+2, area>81: 9(x+2)>81.
- Worked-method reminder: x > 7
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