Class 11 Maths Chapter 5 Linear Inequalities – Exercise 5.3 NCERT Solutions

Introduction

Exercise 5.3 focuses on solving linear inequalities in two variables using graphical methods. Students learn how to represent inequalities on a coordinate plane, identify feasible regions, and interpret solutions. This exercise is crucial for understanding linear programming and optimization problems in higher classes.

Key Concepts

1. Linear Inequality in Two Variables: General form:

$$\mathrm{<span\; style="font-family:\; verdana;"><b>a</b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>x</b></span>}<span\; style="font-family:\; verdana;"><b>+</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>b</b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>y</b></span>}<span\; style="font-family:\; verdana;"><b>+</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>c</b></span>}<span\; style="font-family:\; verdana;"><b>\le </b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>0</b></span>}<span\; style="font-family:\; verdana;"><b>,</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>a</b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>x</b></span>}<span\; style="font-family:\; verdana;"><b>+</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>b</b></span>}\mathrm{<span\; style="font-family:\; verdana;"><b>y</b></span>}<span\; style="font-family:\; verdana;"><b>+</b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>c</b></span>}<span\; style="font-family:\; verdana;"><b>\ge </b></span>\mathrm{<span\; style="font-family:\; verdana;"><b>0</b></span>}$$

2. Graphical Representation:

   • Replace inequality with equality to draw boundary line.

   • Use test point to determine which side of line satisfies inequality.

   • Shade feasible region accordingly.

3. Solution Set: All points in shaded region represent solutions of inequality.

Common Mistakes

• Forgetting to test a point to identify correct region.

• Confusing strict inequalities ($<,>$) with non‑strict ($\le ,\ge$).

• Not drawing boundary line correctly.

• Ignoring feasible region when multiple inequalities are given.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Solve graphically: $2x+y\le 10$. Boundary line: $2x+y=10$. Test point $(0,0)$: $0\le 10$ → true. Feasible region: below line including boundary.

Q2. Solve graphically: $x+2y\ge 6$. Boundary line: $x+2y=6$. Test point $(0,0)$: $0\ge 6$ → false. Feasible region: above line including boundary.

Q3. Solve graphically: $3x+4y<12$. Boundary line: $3x+4y=12$. Test point $(0,0)$: $0<12$ → true. Feasible region: below line, boundary not included.

Q4. Solve graphically: $x-y\le 2$. Boundary line: $x-y=2$. Test point $(0,0)$: $0-0\le 2$ → true. Feasible region: below line including boundary.

Q5. Solve graphically: $2x-3y\ge 6$. Boundary line: $2x-3y=6$. Test point $(0,0)$: $0\ge 6$ → false. Feasible region: above line including boundary.

Q6. Solve graphically: $x+y>4$. Boundary line: $x+y=4$. Test point $(0,0)$: $0>4$ → false. Feasible region: above line, boundary not included.

Q7. Solve graphically: $5x+2y\le 20$. Boundary line: $5x+2y=20$. Test point $(0,0)$: $0\le 20$ → true. Feasible region: below line including boundary.

Q8. Solve graphically: $x-2y\ge -4$. Boundary line: $x-2y=-4$. Test point $(0,0)$: $0\ge -4$ → true. Feasible region: above line including boundary.

Q9. Solve graphically: $3x-y<9$. Boundary line: $3x-y=9$. Test point $(0,0)$: $0<9$ → true. Feasible region: below line, boundary not included.

Q10. Solve graphically: $x+3y\le 12$. Boundary line: $x+3y=12$. Test point $(0,0)$: $0\le 12$ → true. Feasible region: below line including boundary.

FAQs (10)

FAQ1. What is linear inequality in two variables? An inequality involving $x$ and $y$.

FAQ2. How to represent inequality graphically? Draw boundary line and shade feasible region.

FAQ3. What is feasible region? Region satisfying inequality conditions.

FAQ4. How to check which side to shade? Use test point like $(0,0)$.

FAQ5. What if inequality is strict? Boundary line is not included.

FAQ6. What if inequality is non‑strict? Boundary line is included.

FAQ7. Why is Exercise 5.3 important? It builds foundation for linear programming.

FAQ8. What is boundary line equation? Replace inequality sign with equality.

FAQ9. Can inequalities have infinite solutions? Yes, all points in feasible region.

FAQ10. What is practical use of inequalities? Used in optimization, economics, and resource allocation.

Conclusion

Exercise 5.3 covers graphical solutions of linear inequalities in two variables. With solved examples and FAQs, students gain clarity on representing inequalities and identifying feasible regions.

visit:www.fuzymathacademy.com

New Syllabus-Class 11 Maths Chapter 5 Linear Inequalities – Exercise 5.3 NCERT Solutions