Linear Inequalities – Class 11 Maths Chapter 5 Notes & NCERT PDF

Class 11MathematicsChapter 5NCERT book: Mathematics - Class 11

Notes and a simple summary of Chapter 5 of the NCERT Class 11 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 5: Linear Inequalities

Introduction

Not every real-life statement is an equation. "Ravi has ₹200 and rice costs ₹30 a packet" gives 30x < 200, and "Reshma has ₹120 for registers at ₹40 and pens at ₹20" gives 40x + 20y ≤ 120. Statements that use <, >, ≤ or ≥ are inequalities. This chapter (rationalised syllabus) studies linear inequalities in one variable — how to solve them algebraically and how to show the solutions on a number line — and applies them to word problems in marks, temperature and mixtures.

Definition 1: Two real numbers or two algebraic expressions related by the symbol <, >, ≤ or ≥ form an inequality.

Key Concepts

5.2 Types of inequalities

Numerical inequalities: 3 < 5, 7 > 5.

Literal inequalities: x < 5, y > 2, x ≥ 3, y ≤ 4.

Double inequalities: 3 < x < 5, 2 < y < 4.

Strict inequalities use < or > (e.g. ax + b < 0); slack inequalities use ≤ or ≥ (e.g. ax + b ≤ 0).

Linear in one variable: ax + b < 0, ax + b > 0, ax + b ≤ 0, ax + b ≥ 0 with a ≠ 0.

Linear in two variables: ax + by < c etc. with a ≠ 0, b ≠ 0.

ax² + bx + c ≤ 0 or > 0 (a ≠ 0) is quadratic, not linear.

5.3 Solving linear inequalities in one variable

A solution is a value of the variable that makes the inequality a true statement; the set of all such values is the solution set. For 30x < 200 with x a count of packets, trial gives the solution set {0, 1, 2, 3, 4, 5, 6} — slow, so we use rules.

Rule 1: Equal numbers may be added to (or subtracted from) both sides without changing the sign of inequality.

Rule 2: Both sides may be multiplied (or divided) by the same positive number. If both sides are multiplied or divided by a negative number, the sign of inequality is reversed (< becomes >, ≤ becomes ≥).

Why the reversal? 3 > 2 but −3 < −2; and −8 < −7 but (−8)(−2) = 16 > 14 = (−7)(−2).

The universe matters. The same inequality has different solution sets when x must be a natural number, an integer or a real number. Unless stated otherwise, solve over the real numbers and write the answer in interval notation.

Graphical representation on the number line

For x < a or x > a (strict): draw an open (hollow) circle at a and darken the line to the left or right.

For x ≤ a or x ≥ a (slack): draw a filled (dark) circle at a and darken the line to the left or right.

Worked Examples (from the chapter)

Example 1: 30x < 200 ⇒ x < 20/3. If x is a natural number: {1, 2, 3, 4, 5, 6}. If x is an integer: {…, −3, −2, −1, 0, 1, 2, …, 6}.

Example 3: 4x + 3 < 6x + 7 ⇒ −2x < 4 ⇒ x > −2 (divided by −2, sign reversed). Solution set (−2, ∞).

Example 4: (5 − 2x)/3 ≤ x/6 − 5. Multiply by 6: 2(5 − 2x) ≤ x − 30 ⇒ 10 − 4x ≤ x − 30 ⇒ −5x ≤ −40 ⇒ x ≥ 8, i.e. x ∈ [8, ∞).

Example 7 (marks): Terminal marks 62 and 48; average at least 60. (62 + 48 + x)/3 ≥ 60 ⇒ 110 + x ≥ 180 ⇒ x ≥ 70. Minimum annual marks = 70.

Example 8: Consecutive odd natural numbers, both > 10, sum < 40. With x > 10 and 2x + 2 < 40 ⇒ 10 < x < 19, so x = 11, 13, 15, 17 → pairs (11, 13), (13, 15), (15, 17), (17, 19).

Example 11 (system): 3x − 7 < 5 + x gives x < 6; 11 − 5x ≤ 1 gives x ≥ 2. Common part: 2 ≤ x < 6 (filled circle at 2, open circle at 6).

Example 12 (temperature): 30 < C < 35 with C = (5/9)(F − 32) ⇒ 54 < F − 32 < 63 ⇒ 86 < F < 95 (°F).

Example 13 (mixture): 600 L of 12% acid; add x L of 30% acid so that the result is between 15% and 18%. 30x + 7200 > 15x + 9000 ⇒ x > 120 and 30x + 7200 < 18x + 10800 ⇒ x < 300. So 120 < x < 300 litres.

Formula & Rule Sheet

Adding or subtracting the same number: sign unchanged.

Multiplying or dividing by a positive number: sign unchanged.

Multiplying or dividing by a negative number: sign reversed.

Double inequality a < px + q < b: apply the same operation to all three parts.

Interval notation: x < a ↔ (−∞, a); x ≤ a ↔ (−∞, a]; x > a ↔ (a, ∞); x ≥ a ↔ [a, ∞).

System of inequalities: solve each, then take the common part (intersection).

Exam Focus

Exercise 5.1, Q1–4: solve over natural numbers, integers or reals — list elements for N and Z, use intervals for R.

Exercise 5.1, Q5–16: clear fractions by multiplying by the LCM (positive), then collect terms.

Q17–20: solve and draw on the number line; state open or filled circle.

Word problems (Q21–26): define the variable, write the inequality, solve, then interpret (e.g. "at least" → ≥, "more than" → >).

Answer outline — "Solve 5x − 3 < 7 when x is an integer": 5x < 10 ⇒ x < 2 ⇒ solution set {…, −2, −1, 0, 1}.

Answer outline — "Solve −8 ≤ 5x − 3 < 7": add 3 → −5 ≤ 5x < 10; divide by 5 → −1 ≤ x < 2; interval [−1, 2).

Key Terms

Inequality (असमिका) — a statement using <, >, ≤ or ≥.

Linear inequality (रैखिक असमिका) — highest power of the variable is 1.

Strict inequality (निरंकुश असमिका) — uses < or >.

Slack inequality (शिथिल असमिका) — uses ≤ or ≥.

Solution (हल) — a value that makes the inequality true.

Solution set (हल समुच्चय) — the set of all solutions.

Double inequality (दोहरी असमिका) — e.g. 3 < x < 5.

Number line (संख्या रेखा) — used to show solutions graphically.

Interval (अंतराल) — (a, b), [a, b), etc.

System of inequalities (असमिकाओं का निकाय) — solved together; answer is the common part.

Open circle (खुला वृत्त) — endpoint not included.

Closed/filled circle (भरा वृत्त) — endpoint included.

Common Mistakes

Forgetting to reverse the sign after multiplying or dividing by a negative number (e.g. writing −2x < 4 ⇒ x < −2).

Ignoring the universe: giving an interval when the question says x is a natural number.

Using a filled circle for a strict inequality, or an open circle for ≤ / ≥.

In a system, giving the union instead of the intersection of the solution sets.

Translating "at least" as > instead of ≥, or "at most" as < instead of ≤.

Multiplying by a variable expression whose sign is unknown — only multiply by known positive or negative numbers.

💡 Key Learning Points

✓Recognise numerical, literal, double, strict and slack inequalities and identify which are linear.
✓Solve linear inequalities in one variable using Rule 1 (add/subtract) and Rule 2 (multiply/divide, reversing the sign for negatives).
✓State solution sets correctly for natural numbers, integers and real numbers, using interval notation for reals.
✓Represent solutions on a number line with open circles for strict and filled circles for slack inequalities.
✓Model and solve word problems (marks, consecutive numbers, temperature conversion, acid mixtures) with inequalities and systems.

👨‍🏫 Teaching Tips

→Open with the chapter's own situations (Ravi's ₹200 rice packets, Reshma's registers and pens) to show why equations are not enough.
→Demonstrate the sign reversal numerically (3 > 2 but −3 < −2) before stating Rule 2 formally.
→Solve one inequality three times — over N, Z and R — to make the role of the universe visible.
→Use colour for number-line graphs: hollow dot vs filled dot, and shade the common part of a system in a third colour.
→Set a 'translate the phrase' drill: at least, at most, more than, less than, between, and map each to a symbol.
→Pair students to check each other's work specifically for a missed sign reversal.
→Close with the mixture problem (Example 13) as a real-world application needing two inequalities.

📋 Assessment Questions

Can the student solve ax + b < c when a is negative and correctly reverse the sign?
Can the student write the solution set over N or Z as a list and over R as an interval?
Can the student draw a solution on the number line with the correct open or filled endpoint?
Can the student solve a system of two inequalities and state the common solution?
Can the student form and solve an inequality from a word problem and interpret the answer?

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