Ncert Books Class 11 Maths Chapter 6 Free Download

To correctly solve linear inequalities and score full marks, students must master these two fundamental rules:

• Rule 1: Equal numbers can be added to or subtracted from both sides of an inequality without changing the sign of inequality.
• Rule 2: Both sides of an inequality can be multiplied or divided by the same positive number. However, when you multiply or divide by a negative number, the sign of inequality is reversed (i.e., ‘<’ becomes ‘>’, and ‘≥’ becomes ‘≤’). Misapplying this rule is a very common source of errors in exams.

Reversing the inequality sign is a mathematical necessity to maintain the truth of the statement. For instance, we know that 4 is greater than 2 (4 > 2). If you multiply both sides by -1 without reversing the sign, you get -4 > -2, which is false. To keep the statement true, you must reverse the sign to get -4 < -2. In exams, questions are often designed to test this specific concept, and failing to reverse the sign leads to a completely incorrect solution set, resulting in zero marks for that step.

Solving a system of linear inequalities graphically is a high-yield topic. Follow these steps for a complete answer:

1. Step 1: Treat each inequality as an equation (e.g., change ax + by > c to ax + by = c).
2. Step 2: Draw the line for each equation on the Cartesian plane. Use a solid line for inequalities with ≤ or ≥ and a dotted line for inequalities with < or >.
3. Step 3: For each inequality, determine the solution region. Pick a test point (like (0,0)) not on the line. If it satisfies the inequality, shade the region containing the test point; otherwise, shade the other region.
4. Step 4: The common shaded area among all inequalities is the final feasible solution region for the system. Clearly label this region in your answer.

To maximise your score, be careful to avoid these common pitfalls:

• Using Solid vs. Dotted Lines: Using a solid line for a strict inequality (< or >) or a dotted line for a non-strict inequality (≤ or ≥) is a frequent error that can cost marks.
• Incorrect Shading: Shading the wrong side of the line is the most critical error. Always use a test point like (0,0) to confirm the correct region.
• Forgetting to Shade: The solution to a graphical inequality is the shaded region, not just the line. Forgetting to shade or failing to clearly indicate the final feasible region for a system is a major mistake.
• Ignoring Constraints: In word problems, implicit constraints like x ≥ 0 and y ≥ 0 are often present (e.g., number of items cannot be negative). Forgetting to include these in your graph will lead to an incorrect solution region.

Based on CBSE exam trends for the 2025-26 syllabus, the most important areas to focus on in Linear Inequalities are:

• Graphical Solution of a System of Linear Inequalities in Two Variables: This is the highest-weightage topic, often appearing as a 3, 4, or 5-mark question.
• Word Problems: The ability to formulate linear inequalities from a real-world scenario (e.g., manufacturing, diet, or allocation problems) and then solve them graphically is a key skill tested in long-answer questions.
• Algebraic Solution of Inequalities in One Variable: While simpler, questions requiring representation on a number line are very common in the short-answer section.

Word problems are extremely important as they are considered Higher Order Thinking Skills (HOTS) questions. They test a student's ability to apply mathematical concepts to real-world situations. In an exam, these questions carry significant weightage (often 5-6 marks) and differentiate top-performing students. The key skills tested are:

• Translating a descriptive problem into a mathematical system of inequalities.
• Identifying all constraints, including implicit ones (like non-negativity).
• Solving the system graphically to find the feasible region.
• Interpreting the solution in the context of the original problem.