Linear Inequalities
Solving problems with constraints — when equations aren't enough, inequalities show the range of possibilities
In this chapter
5.1 What are Inequalities?
"Mathematics is the art of saying many things in many different ways." — Maxwell
In earlier classes, you learned to solve equations. But what about statements like "the height of all students is less than 160 cm" or "the classroom can hold at most 60 tables"? These use inequality symbols instead of equals signs.
Inequality Symbols
Less Than
a < b means a is strictly less than b. Example: 3 < 5
Greater Than
a > b means a is strictly greater than b. Example: 7 > 5
Less Than or Equal
a ≤ b means a is less than or equal to b. Example: x ≤ 4
Greater Than or Equal
a ≥ b means a is greater than or equal to b. Example: x ≥ 3
Types of Inequalities
Numerical inequalities: 3 < 5, 7 > 5
Literal inequalities: x < 5, y > 2, x ≥ 3
Double inequalities: 3 < x < 5 (x is between 3 and 5)
Linear inequalities in one variable: ax + b < 0, ax + b ≥ 0
Linear inequalities in two variables: ax + by < c, ax + by ≥ c
Strict inequalities: Use < or > (no equality)
Slack inequalities: Use ≤ or ≥ (includes equality)
• Ravi has ₹200 and rice costs ₹30/kg: 30x < 200 (where x = kg bought)
• Reshma has ₹120 for registers (₹40) and pens (₹20): 40x + 20y ≤ 120
• Temperature must stay between 30°C and 35°C: 30 < C < 35
5.2 Rules for Solving Inequalities
Solving inequalities is similar to solving equations, but with one critical difference when multiplying or dividing by negative numbers.
Rule 1: Addition and Subtraction
You can add or subtract the same number from both sides without changing the inequality sign.
If x > 3, then x + 5 > 3 + 5, so x + 5 > 8
If x ≤ 7, then x - 2 ≤ 7 - 2, so x - 2 ≤ 5
Rule 2: Multiplication and Division
By a POSITIVE number: Sign stays the same
If x > 3, then 2x > 6 ✓
By a NEGATIVE number: Sign REVERSES!
If x > 3, then -2x < -6 (sign flipped from > to <)
Why? Because 3 > 2, but -3 < -2. Multiplying by negative flips the order.
Think of it like a seesaw: multiplying by a negative number flips the seesaw, so the heavier side goes up instead of down!
5.3 Solving Linear Inequalities
Now let's solve some linear inequalities step by step.
Example 1: Simple Inequality
Step 1: Divide both sides by 30 (positive, so sign stays):
x < 200/30
x < 20/3 ≈ 6.67
Solution: All real numbers less than 20/3
Interval notation: x ∈ (-∞, 20/3)
Example 2: Two-Step Inequality
Step 1: Subtract 3x from both sides:
2x - 3 < 1
Step 2: Add 3 to both sides:
2x < 4
Step 3: Divide by 2:
x < 2
Solution: x ∈ (-∞, 2)
Example 3: Negative Coefficient
Step 1: Subtract 6x from both sides:
-2x + 3 < 7
Step 2: Subtract 3:
-2x < 4
Step 3: Divide by -2 (REVERSE the sign!):
x > -2
Solution: x ∈ (-2, ∞)
Example 4: Fractions
Step 1: Cross-multiply (both denominators positive):
5(5-2x) ≤ 3(x-6)
Step 2: Expand:
25 - 10x ≤ 3x - 18
Step 3: Combine like terms:
-13x ≤ -43
Step 4: Divide by -13 (REVERSE sign):
x ≥ 43/13 ≈ 3.31
Solution: x ∈ [43/13, ∞)
Double Inequalities
This is two inequalities combined:
-8 ≤ 5x - 3 AND 5x - 3 < 7
Solve both simultaneously:
-8 + 3 ≤ 5x < 7 + 3
-5 ≤ 5x < 10
-1 ≤ x < 2
Solution: x ∈ [-1, 2)
5.4 Graphical Representation
Inequalities can be shown on a number line using circles and shading.
How to Graph on Number Line
Open Circle
Use for < or > (strict inequality). The number itself is NOT included.
Closed Circle
Use for ≤ or ≥ (slack inequality). The number itself IS included.
x < a or x ≤ a: Shade LEFT of a (smaller numbers)
x > a or x ≥ a: Shade RIGHT of a (larger numbers)
Examples
| Inequality | Circle | Shade |
|---|---|---|
| x < 3 | Open at 3 | Left |
| x ≥ -2 | Closed at -2 | Right |
| -1 ≤ x < 5 | Closed at -1, Open at 5 | Between |
5.5 Word Problems
Inequalities are powerful tools for solving real-world problems with constraints.
Example: Minimum Marks Problem
A student scored 62 and 48 in two tests. What's the minimum score needed in the third test to average at least 60?
Solution:
Let x = marks in third test
Average ≥ 60
(62 + 48 + x)/3 ≥ 60
110 + x ≥ 180
x ≥ 70
Answer: Minimum 70 marks needed
Example: Consecutive Odd Numbers
Find all pairs of consecutive odd natural numbers, both > 10, with sum < 40.
Solution:
Let x = smaller odd number, then x + 2 = larger
x > 10 AND x + (x+2) < 40
x > 10 AND 2x + 2 < 40
x > 10 AND x < 19
So 10 < x < 19. Odd numbers in this range: 11, 13, 15, 17
Answer: Pairs are (11,13), (13,15), (15,17), (17,19)
Example: Temperature Conversion
Keep a solution between 30°C and 35°C. What's the range in Fahrenheit?
Formula: C = (5/9)(F - 32)
Solution:
30 < C < 35
30 < (5/9)(F - 32) < 35
54 < F - 32 < 63
86 < F < 95
Answer: Between 86°F and 95°F
Example: Mixture Problem
600L of 12% acid solution. How much 30% solution to add so mixture is 15-18% acid?
Solution:
Let x = liters of 30% solution
Total mixture = x + 600 liters
For > 15%: 0.30x + 0.12(600) > 0.15(x + 600)
→ 30x + 7200 > 15x + 9000 → 15x > 1800 → x > 120
For < 18%: 0.30x + 0.12(600) < 0.18(x + 600)
→ 30x + 7200 < 18x + 10800 → 12x < 3600 → x < 300
Answer: 120 < x < 300 liters
5.6 Summary
• Inequality symbols: < (less), > (greater), ≤ (less or equal), ≥ (greater or equal)
• Rule 1: Add/subtract same number on both sides — sign unchanged
• Rule 2: Multiply/divide by positive — sign unchanged. By negative — sign REVERSES
• Graphing: Open circle for < or >, closed circle for ≤ or ≥
• Shading: Left for less than, right for greater than
• Double inequality: Solve both parts simultaneously
• Word problems: Translate constraints into inequalities, then solve