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JAC • Class XI • Mathematics • Ch 5
Estimated Time: 45 Mins
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Linear Inequalities

In Class 11 Mathematics, "Linear Inequalities" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

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Overview Core Concepts Summary Self Check Take MCQ Test

⚖️ Have You Ever Wondered?

In real-world engineering and business, you rarely have exact equalities like 'Cost = $1,000'; instead, you have constraints like 'Budget $\le$ $10,00...

In real-world engineering and business, you rarely have exact equalities like 'Cost = $1,000'; instead, you have constraints like 'Budget $\le$ $10,000$' and 'Load capacity $\ge$ 500 tons'. Linear inequalities define feasible regions of optimization.

Why This Chapter Matters

In Class 11 Mathematics, "Linear Inequalities" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.

Before You Begin (Prerequisites)

  • Linear equations in one and two variables.
  • Number line representations.
  • Coordinate geometry.

What You Will Learn (Core Objectives)

  • Solve linear inequalities in one variable algebraically and represent solutions on the real number line.
  • Understand reversal of inequality signs when multiplying or dividing by negative numbers.
  • Solve systems of linear inequalities in one variable and find intersection solution intervals.
  • Graph linear inequalities in two variables using boundary lines and half-planes.
  • Determine the feasible solution region for a system of simultaneous linear inequalities graphically.

Chapter Roadmap & Progression

1 1. Algebraic Solutions in One Varia...
2 2. Graphical Solutions in Two Varia...
3 3. Systems of Inequalities

Complete Concept Guide (100% Curriculum Coverage)

1. Algebraic Solutions in One Variable

Two real numbers or algebraic expressions related by $<, >, \le,$ or $\ge$ form an Inequality.
• Cardinal Golden Rule: If both sides are multiplied or divided by a negative number, the inequality sign MUST be reversed! $$\mathbf{-2x < 6 \implies x > -3}$$ Solutions are expressed as intervals: $(a, b)$ open, $[a, b]$ closed.

2. Graphical Solutions in Two Variables

The equation $ax + by = c$ divides the Cartesian plane into two half-planes. To determine which half-plane satisfies $ax + by < c$, test the origin $(0, 0)$: if $(0, 0)$ satisfies the inequality, shade the half-plane containing the origin; otherwise, shade the opposite side! (Dashed boundary line for strict $<, >$; solid line for $\le, \ge$).

3. Systems of Inequalities

The solution region of a system of simultaneous inequalities is the common intersecting shaded region satisfying all given constraints simultaneously.

Visual Learning & Conceptual Map

Linear Inequalities Master Matrix

Conceptual framework, core mechanisms, and analytical relationships
Academic Architecture

1. Algebraic Solutions in One Variable • 2. Graphical Solutions in Two Variables

Chapter Summary & 10 Key Takeaways

Takeaway 1
Golden Rule: Multiplying or dividing by a negative number inverts the inequality sign.
Takeaway 2
Interval Notation: Brackets $[a, b]$ indicate inclusion; parentheses $(a, b)$ indicate exclusion.
Takeaway 3
Half-Plane: Region of Cartesian plane bounded by a line satisfying an inequality.
Takeaway 4
Origin Test: Plugging $(0, 0)$ into an inequality to identify the correct shaded half-plane.
Takeaway 5
Feasible Region: Intersection of multiple half-planes satisfying all constraints.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Solve $30x < 200$ when: (i) $x$ is a natural number, (ii) $x$ is an integer.
Reveal Answer & Explanation
Answer: $x < \frac{200}{30} \implies x < 6.67$. (i) If $x \in \mathbb{N}$, solution set is $\{1, 2, 3, 4, 5, 6\}$. (ii) If $x \in \mathbb{Z}$, solution set is $\{\dots, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}$.
(i) {1,2,3,4,5,6}; (ii) {..., 0, 1, ..., 6}.
2
Solve the inequality: $\frac{3(x-2)}{5} \le \frac{5(2-x)}{3}$.
Reveal Answer & Explanation
Answer: Multiply both sides by 15: $9(x - 2) \le 25(2 - x) \implies 9x - 18 \le 50 - 25x \implies 34x \le 68 \implies x \le 2$. Solution interval is $(-\infty, 2]$.
x ∈ (-∞, 2].
3
Solve $-12x > 30$ when $x$ is a real number.
Reveal Answer & Explanation
Answer: Divide both sides by $-12$ and reverse the inequality sign: $x < -\frac{30}{12} \implies x < -\frac{5}{2}$. Solution interval is $(-\infty, -2.5)$.
x ∈ (-∞, -2.5).
4
How do you determine whether a boundary line should be drawn solid or dashed when graphing an inequality in two variables?
Reveal Answer & Explanation
Answer: If the inequality includes equality ($\le$ or $\ge$), the points on the boundary line are included in the solution set and the line is drawn solid; if strict ($<$ or $>$), the line is dashed.
Solid for ≤ and ≥; dashed for < and >.
5
Find all pairs of consecutive odd positive integers, both of which are smaller than 10, such that their sum is more than 11.
Reveal Answer & Explanation
Answer: Let consecutive odd integers be $x$ and $x + 2$. Constraints: $x < 10$, $x+2 < 10 \implies x < 8$. Sum: $x + (x+2) > 11 \implies 2x + 2 > 11 \implies 2x > 9 \implies x > 4.5$. Odd integers for $x$ are 5 and 7. The pairs are $(5, 7)$ and $(7, 9)$.
Pairs are (5, 7) and (7, 9).
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