Quick Revision Notes: Class 11 Mathematics

Chapter: Linear Inequalities (रैखिक असमिकाएँ)

1. Introduction & Basic Definitions

• Inequality (असमिका): A statement or relation involving a symbol of inequality (<, >, <=, >=) between two real numbers or two algebraic expressions.
• Linear Inequality in One Variable: An inequality that can be written in the form ax + b < 0 (or <=, >, >=), where a != 0 and x is a variable. Example: 2x - 5 > 0.
• Linear Inequality in Two Variables: An inequality that can be written in the form ax + by < c (or <=, >, >=), where a, b != 0 and x, y are variables. Example: 3x + 4y <= 12.
• Solution of an Inequality: The value(s) of the variable(s) which make the inequality a true statement.
• Solution Set: The set of all solutions of an inequality.

2. Rules for Solving Algebraic Linear Inequalities

When solving inequalities in one variable, we can perform the following operations without changing the inequality sign:

1. Addition/Subtraction: Equal numbers can be added to (or subtracted from) both sides of an inequality.
   • If x < y, then x + c < y + c and x - c < y - c.
2. Multiplication/Division by Positive Number: Both sides of an inequality can be multiplied (or divided) by the same positive number.
   • If x < y and c > 0, then cx < cy and x/c < y/c.
3. Multiplication/Division by Negative Number: When both sides are multiplied (or divided) by a negative number, the direction of the inequality sign is reversed.
   • If x < y and c < 0, then cx > cy and x/c > y/c.

3. Representation of Intervals on Real Number Line

Intervals are subsets of real numbers and are represented as follows:

• Closed Interval [a, b]: Includes all real numbers between a and b, including a and b.
  • Set builder: {x : a <= x <= b}
• Open Interval (a, b): Includes all real numbers between a and b, excluding a and b.
  • Set builder: {x : a < x < b}
• Semi-Open / Semi-Closed Intervals:
  • [a, b): {x : a <= x < b} (Includes a, excludes b)
  • (a, b]: {x : a < x <= b} (Excludes a, includes b)
• Infinite Intervals:
  • [a, infinity): {x : x >= a}
  • (-infinity, a): {x : x < a}

4. Graphical Solution of Linear Inequalities in Two Variables

To solve ax + by <= c (or other inequality symbols) graphically:

1. Step 1: Replace the inequality sign with an equality sign (=) to get the boundary line equation: ax + by = c.
2. Step 2: Draw the corresponding graph (line).
   • Use a solid line (—) for inequalities involving <= or >= (points on the line are included).
   • Use a dashed line (---) for inequalities involving < or > (points on the line are not included).
3. Step 3: Choose a test point (usually origin (0, 0) if it doesn't lie on the line) and substitute its coordinates into the inequality.
4. Step 4:
   • If the test point satisfies the inequality, shade the half-plane containing the test point.
   • If it does not satisfy, shade the half-plane not containing the test point.

5. Solution of a System of Linear Inequalities

• When two or more linear inequalities are considered simultaneously, they form a system of linear inequalities.
• The solution region (or feasible region) of a system of inequalities is the intersection (common region) of the solution regions of all individual inequalities in the system.
• If there is no common region, the system has no solution (inconsistent).