5.1 Inequalities — Introduction and Types
An inequality is formed when two real numbers or algebraic expressions are related by the symbols <, >, ≤ or ≥. Unlike equations, inequalities give a range of solutions rather than a specific value.
Numerical: 3 < 5, 7 > 5 (comparing two numbers)
Literal: x < 5, y ≥ 2 (involving variables)
Double: 3 ≤ x < 5 (variable between two values)
Strict inequalities: use < or > (excludes equality)
Slack inequalities: use ≤ or ≥ (includes equality)
[ Image: Number line showing different types of inequality representations — add here ]
Classification by number of variables:
Linear in one variable: ax + b < 0, ax + b ≥ 0
Example: 5x – 3 < 7
Linear in two variables: ax + by < c, ax + by ≥ c
Example: 40x + 20y ≤ 120
Quadratic (NOT in scope): ax² + bx + c ≤ 0
We study only linear inequalities in this chapter.
Solution of an inequality: Any value of the variable that makes the inequality a true statement. The set of all such values is the solution set.
🎯 Practice MCQs
Slack inequalities use ≤ or ≥ (they include the equality case). Option C (ax + by ≤ c) uses ≤, so it is a slack inequality. Options A and B use strict symbols (<, >).
Since rice is available only in whole packets and Ravi cannot spend exactly ₹200 in all cases (200/30 = 6.67 is not an integer), the total amount spent 30x must be strictly less than 200. So 30x < 200.
30x < 200 → x < 200/30 = 6.67. Natural numbers less than 6.67 are 1, 2, 3, 4, 5, 6. (Note: 0 is NOT a natural number — natural numbers start from 1.)
5.2 Solving Linear Inequalities in One Variable
The two rules for solving inequalities are similar to equations — with one critical difference:
Equal numbers may be added to or subtracted from both sides of an inequality without changing the inequality sign.
If a > b, then a + c > b + c and a – c > b – c (for any real c).
Both sides of an inequality can be multiplied or divided by the same positive number without changing the sign.
BUT: When multiplying or dividing by a NEGATIVE number, the inequality sign is REVERSED!
3 > 2 → –3 < –2 (sign reversed when multiplied by –1)
Critical Rule: Multiplying or dividing both sides by a negative number reverses the inequality sign. < becomes >, ≤ becomes ≥, and vice versa. This is the most common source of errors!
[ Image: Number line graphs for x < 3 (open circle) and x ≥ 1 (filled circle) — add here ]
Solution sets and their interval notation:
x < 2 → solution: (–∞, 2) — open circle at 2, arrow pointing left
x ≥ –2 → solution: [–2, ∞) — filled circle at –2, arrow pointing right
–1 ≤ x < 2 → solution: [–1, 2) — filled at –1, open at 2
🎯 Practice MCQs
5x – 3 < 3x + 1
5x – 3x < 1 + 3 (adding 3 and subtracting 3x from both sides)
2x < 4 → x < 2.
Solution: x ∈ (–∞, 2).
4x + 3 < 6x + 7
4x – 6x < 7 – 3
–2x < 4
Dividing by –2 (negative) → sign reverses: x > –2.
Solution: x ∈ (–2, ∞).
(5–2x)/3 ≤ x/6 – 5
Multiply both sides by 6:
2(5–2x) ≤ x – 30
10 – 4x ≤ x – 30
–5x ≤ –40
Divide by –5 (sign reverses): x ≥ 8. Solution: x ∈ [8, ∞).
5.3 System of Inequalities and Real-Life Applications
A system of inequalities means solving two or more inequalities simultaneously. The solution is the set of values that satisfy all the inequalities at the same time (intersection of individual solution sets).
An inequality of the form a ≤ expr < b means solving two inequalities together:
a ≤ expr AND expr < b.
Example: –8 ≤ 5x – 3 < 7
→ Add 3: –5 ≤ 5x < 10
→ Divide by 5: –1 ≤ x < 2 → solution: [–1, 2)
Solve each inequality separately, then take the common (intersecting) solution.
Example: 3x – 7 < 5 + x → x < 6
AND 11 – 5x ≤ 1 → x ≥ 2
Intersection: 2 ≤ x < 6
[ Image: Number line showing intersection of two solution sets (bold overlap region) — add here ]
Temperature Conversion Application (NCERT Example 12):
Solution must stay between 30°C and 35°C. Using C = 5/9 × (F–32):
30 < 5/9(F–32) < 35 → multiply by 9/5 → 54 < F–32 < 63 → 86 < F < 95.
Acid mixture: 600 L of 12% + x L of 30% must give 15% to 18%.
Solving both inequalities simultaneously: 120 < x < 300.
Consecutive Number Problems: If x is the smaller of two consecutive odd numbers, the other is x+2. Set up system of inequalities and find the valid integer values.
🎯 Practice MCQs
–8 ≤ 5x – 3 < 7
Add 3 throughout: –8+3 ≤ 5x < 7+3
–5 ≤ 5x < 10
Divide by 5: –1 ≤ x < 2 → x ∈ [–1, 2).
Let x = marks in 3rd test.
(62 + 48 + x)/3 ≥ 60
(110 + x)/3 ≥ 60
110 + x ≥ 180
x ≥ 70. Minimum marks needed = 70.
30 < C < 35. Substituting C = 5(F–32)/9:
30 < 5(F–32)/9 < 35
Multiply by 9/5: 54 < F–32 < 63
Add 32: 86 < F < 95.
Range: 86°F to 95°F.
🚀 Clear Linear Inequalities for AAI ATC!
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