1. Chapter at a Glance
Not every question asks "how much exactly?". Often we ask "at least how many?" or "at most how much?". A student needs at least 60 marks on average; a lift can carry at most 500 kg; a shopkeeper's profit must exceed ₹5,000. Such conditions are written as inequalities using <, >, ≤ and ≥. In this chapter we solve linear inequalities in one variable algebraically, show their solution sets on the number line and as intervals, and apply them to word problems.
- Strict and slack inequalities; linear inequalities in one variable.
- Rules for solving: adding, subtracting, multiplying, dividing.
- Solution sets over ℕ, ℤ and ℝ.
- Representing solutions on the number line and in interval notation.
- Systems (simultaneous) inequalities and double inequalities.
- Word problems: marks, costs, temperatures, solutions (mixtures).
2. Inequalities
Two expressions compared with <, >, ≤ or ≥ form an inequality. < and > are strict; ≤ and ≥ are slack. A linear inequality in one variable has the form ax + b < 0 (or >, ≤, ≥), with a ≠ 0.
3. Rules for Solving
- Rule 1: The same number may be added to or subtracted from both sides — the sign stays.
- Rule 2: Both sides may be multiplied or divided by the same positive number — the sign stays.
- Rule 3: When multiplying or dividing both sides by a negative number, the sign reverses (< becomes >, ≤ becomes ≥).
Why rule 3? 2 < 5, but −2 > −5: multiplying by −1 flips the order on the number line.
Solve 30x < 200 when x is (a) a natural number, (b) an integer, (c) a real number.
Solution: x < 20/3 ≈ 6.67. (a) {1, 2, 3, 4, 5, 6}. (b) {…, −2, −1, 0, 1, …, 6}. (c) (−∞, 20/3).
Solve 4x + 3 < 6x + 7.
Solution: −2x < 4 → divide by −2 and reverse: x > −2, i.e. (−2, ∞).
Solve (5 − 2x)/3 ≤ x/6 − 5.
Solution: Multiply by 6: 2(5 − 2x) ≤ x − 30 → 10 − 4x ≤ x − 30 → −5x ≤ −40 → x ≥ 8, i.e. [8, ∞).
4. Number Line and Intervals
| Inequality | Interval | Number line |
|---|---|---|
| x > a | (a, ∞) | Open circle at a, arrow right |
| x ≥ a | [a, ∞) | Filled dot at a, arrow right |
| x < a | (−∞, a) | Open circle, arrow left |
| a ≤ x < b | [a, b) | Filled at a, open at b |
∞ is not a number, so it always has a round bracket.
5. Systems and Double Inequalities
For a system, solve each inequality and take the common part (intersection) of the solution sets.
Solve 3x − 7 < 5 + x and 11 − 5x ≤ 1.
Solution: First: 2x < 12 → x < 6. Second: −5x ≤ −10 → x ≥ 2. Common: [2, 6).
Solve −8 ≤ 5x − 3 < 7.
Solution: Add 3: −5 ≤ 5x < 10 → divide by 5: −1 ≤ x < 2, i.e. [−1, 2).
Solve −3 ≤ 4 − 7x/2 ≤ 18.
Solution: Subtract 4: −7 ≤ −7x/2 ≤ 14. Multiply by −2/7 (reverse both signs): 2 ≥ x ≥ −4 → [−4, 2].
6. Word Problems
Ravi scored 70 and 75 in the first two unit tests. What is the minimum mark he needs in the third so that his average is at least 60?
Solution: (70 + 75 + x)/3 ≥ 60 → x ≥ 35. Minimum 35.
A solution must be kept between 68°F and 77°F. What is the range in °C? (F = 9C/5 + 32)
Solution: 68 < 9C/5 + 32 < 77 → 36 < 9C/5 < 45 → 20 < C < 25.
How many litres of 2% acid solution must be added to 640 L of 8% solution so that the mixture is more than 4% but less than 6% acid?
Solution: Acid = 0.02x + 51.2 in (640 + x) L. 4% < … < 6%: 0.04(640 + x) < 0.02x + 51.2 < 0.06(640 + x). Left: 25.6 + 0.04x < 0.02x + 51.2 → x < 1280. Right: 0.02x + 51.2 < 38.4 + 0.06x → x > 320. 320 < x < 1280 litres.
7. Solved Problems (Board Pattern)
Find all pairs of consecutive odd positive integers, both smaller than 10, whose sum is more than 11.
Solution: x + (x + 2) > 11 → x > 4.5, x + 2 < 10 → x < 8, x odd → x = 5, 7. Pairs (5, 7), (7, 9).
The longest side of a triangle is 3 times the shortest, and the third side is 2 cm shorter than the longest. The perimeter is at least 61 cm. Find the minimum length of the shortest side.
Solution: x + 3x + (3x − 2) ≥ 61 → 7x ≥ 63 → x ≥ 9 cm.
IQ = (MA/CA) × 100. For 12-year-olds, IQ ranges over 80 ≤ IQ ≤ 140. Find the range of mental age (MA).
Solution: 80 ≤ 100MA/12 ≤ 140 → 9.6 ≤ MA ≤ 16.8.
8. Master Formula Table
| Step | Effect on sign |
|---|---|
| Add/subtract any number | No change |
| Multiply/divide by positive | No change |
| Multiply/divide by negative | Reverse |
| ax + b > 0, a > 0 | x > −b/a |
| ax + b > 0, a < 0 | x < −b/a |
| System | Intersection of solution sets |
| Average ≥ k for n tests | sum ≥ nk |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Solving | Like equations, but watch negatives | Linear inequalities with fractions |
| Number sets | ℕ, ℤ, ℝ give different answers | List solutions |
| Representation | Open/closed circles; intervals | Draw on number line |
| Systems | Common region | Two inequalities together |
| Applications | "At least", "at most", "between" | Marks, mixtures, temperatures |
10. Mind Map
- <, > strict
- ≤, ≥ slack
- Negative flips sign
- Clear fractions
- Number line
- Intervals
- Systems
- Word problems
11. Tips and Tricks
- To avoid sign flips, move the variable terms to the side where their coefficient is positive.
- Translate words: "at least" → ≥, "at most" → ≤, "more than" → >, "less than" → <.
- Check your answer with a test value from the solution interval.
- Forgetting to reverse the sign after dividing by a negative number.
- Multiplying both sides by an expression like x whose sign is unknown.
- Using a square bracket with ∞.
12. Important One-liners
- Multiplying or dividing by a negative number reverses an inequality.
- A linear inequality over ℝ has infinitely many solutions (an interval).
- Open circle: endpoint excluded; filled dot: endpoint included.
- The solution of a system is the intersection of individual solutions.
- "At least" means ≥.
13. Exam-pattern Questions with Detailed Answers
Solve −12x > 30 when x is a natural number.
- {1, 2}
- No solution
- {−3, −4, …}
- All natural numbers
Show answer
x < −5/2, and no natural number is negative.
Answer: (B)
Solve 3x + 8 > 2 for real x.
Show answer
3x > −6 → x > −2: (−2, ∞).
Answer: (−2, ∞)
Write −3 < x ≤ 5 in interval notation.
Show answer
(−3, 5].
Answer: (−3, 5]
Solve 3(x − 1) ≤ 2(x − 3).
Show answer
3x − 3 ≤ 2x − 6 → x ≤ −3: (−∞, −3].
Answer: (−∞, −3]
Solve x/3 > x/2 + 1.
Show answer
Multiply by 6: 2x > 3x + 6 → −x > 6 → x < −6.
Answer: (−∞, −6)
Solve the system 5x − 1 < 24 and 5x + 1 > −24.
Show answer
x < 5 and x > −5: (−5, 5).
Answer: (−5, 5)
Solve −15 < 3(x − 2)/5 ≤ 0.
Show answer
Multiply by 5/3: −25 < x − 2 ≤ 0 → −23 < x ≤ 2.
Answer: (−23, 2]
A student must get an average of at least 60 in five tests to pass. Her first four marks are 55, 62, 48 and 70. What must she score in the fifth?
Show answer
235 + x ≥ 300 → x ≥ 65.
Answer: At least 65
Solve (2x − 1)/3 ≥ (3x − 2)/4 − (2 − x)/5.
Show answer
Multiply by 60: 20(2x − 1) ≥ 15(3x − 2) − 12(2 − x) → 40x − 20 ≥ 45x − 30 − 24 + 12x → 40x − 20 ≥ 57x − 54 → −17x ≥ −34 → x ≤ 2.
Answer: (−∞, 2]
A manufacturer has 600 L of a 12% acid solution. How many litres of 30% acid solution must be added so that the mixture is more than 15% but less than 18% acid?
Show answer
Acid = 72 + 0.3x in 600 + x. 0.15(600 + x) < 72 + 0.3x → 90 + 0.15x < 72 + 0.3x → x > 120. 72 + 0.3x < 0.18(600 + x) = 108 + 0.18x → 0.12x < 36 → x < 300.
Answer: 120 < x < 300 litres
Find all pairs of consecutive even positive integers, both larger than 5, whose sum is less than 23.
Show answer
x > 5, x even, 2x + 2 < 23 → x < 10.5. x = 6, 8, 10. Pairs (6, 8), (8, 10), (10, 12).
Answer: (6, 8), (8, 10), (10, 12)
The cost and revenue of producing x units are C(x) = 26000 + 30x and R(x) = 43x. How many units must be sold to make a profit?
Show answer
R > C → 43x > 26000 + 30x → 13x > 26000 → x > 2000.
Answer: More than 2000 units
14. Quick Revision Summary
- Solve inequalities like equations, except: dividing/multiplying by a negative reverses the sign.
- The number set (ℕ, ℤ, ℝ) decides the final solution set.
- Show answers on a number line (open/filled) and as intervals.
- Systems: intersect the solution sets; double inequalities: operate on all three parts.
- Translate "at least / at most / between" carefully.