Section 1: Mindmap

Section 2: Main Content with Inline Diagrams

Introduction to Linear Inequations

A linear inequation is an inequality involving a linear expression in one or more variables. It is similar to a linear equation but instead of an equal sign, it contains inequality signs such as \( <, \leq, >, \geq \).

Example: \( 3x + 5 > 2 \) is a linear inequation in one variable \( x \).

Linear inequations can be classified as:

• One-variable linear inequations: Inequations involving a single variable, e.g., \( 2x - 7 \leq 3 \).
• Two-variable linear inequations: Inequations involving two variables, e.g., \( 3x + 4y \geq 12 \).

Properties of Linear Inequations

When solving linear inequations, certain properties must be remembered:

• You can add or subtract the same number or expression on both sides without changing the inequality.
• You can multiply or divide both sides by a positive number without changing the inequality direction.
• Important: Multiplying or dividing both sides by a negative number reverses the inequality sign.

Solving Linear Inequations in One Variable

To solve a linear inequation like \( ax + b < c \), isolate \( x \) on one side:

Graphical Representation of One-Variable Inequations

The solution set of a linear inequation in one variable is represented on a number line.

• If the inequality is strict (\( < \) or \( > \)), the boundary point is shown as an open circle.
• If the inequality is inclusive (\( \leq \) or \( \geq \)), the boundary point is shown as a closed (filled) circle.
• The region to the left or right of the boundary point is shaded depending on the inequality.

Number line example for \( x \leq 3 \):---●====================>    3

Solving Linear Inequations in Two Variables

Two-variable linear inequations take the form:

The solution set is a region in the coordinate plane bounded by the line \( ax + by = c \).

Steps to graph:

1. Graph the boundary line \( ax + by = c \). Use a solid line if the inequality is inclusive (\( \leq, \geq \)) and a dashed line if strict (\( <, > \)).
2. Choose a test point not on the line (usually \( (0,0) \) if it is not on the line).
3. Substitute the test point into the inequality.
4. If the inequality holds true, shade the region containing the test point; otherwise, shade the opposite side.

Example of Graphical Solution

Inequation: \( 2x + 3y \leq 6 \)Boundary line: \( 2x + 3y = 6 \)- Plot points:   For \( x=0 \), \( y=2 \); For \( y=0 \), \( x=3 \)- Draw solid line through (0,2) and (3,0)- Test point: (0,0)  \( 2(0) + 3(0) = 0 \leq 6 \) (True)- Shade region containing (0,0)

Systems of Linear Inequations

A system of linear inequations consists of two or more inequations considered simultaneously. The solution is the intersection of the solution sets of each inequation.

Example:

Graph each inequation and find the common shaded region where both conditions hold.

Shaded Regions and Feasible Solutions

The common shaded area in the coordinate plane represents all solutions satisfying all inequations in the system. This is important in optimization problems and linear programming.

Set Notations for Solutions

• Interval notation: For one variable, e.g., \( x \in (-\infty, 3] \).
• Set-builder notation: \( { x \mid x \leq 3 } \).
• For two variables, solutions are expressed as sets of ordered pairs satisfying the inequations.

Common Mistakes to Avoid

• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
• Using a solid line for strict inequalities or a dashed line for inclusive inequalities.
• Incorrectly shading the wrong side of the boundary line.
• Not checking the test point properly.

Connection to Board Exams and Previous Year Questions

Questions on linear inequations often appear in NDA and other entrance exams, testing:

• Solving linear inequations algebraically.
• Graphical representation and interpretation of solution sets.
• Systems of inequations and finding common solution regions.

Understanding these concepts thoroughly helps in solving multiple-choice questions and long-answer problems efficiently.

Section 3: Worked Examples

Section 4: Formula Bank

• General form of linear inequation in one variable: \( ax + b < c \), \( ax + b \leq c \), \( ax + b > c \), \( ax + b \geq c \)
• Solution for one variable: \( x < \frac{c - b}{a} \) if \( a > 0 \), else \( x > \frac{c - b}{a} \) if \( a < 0 \) (reverse inequality)
• General form in two variables: \( ax + by < c \), \( ax + by \leq c \), \( ax + by > c \), \( ax + by \geq c \)
• Boundary line equation: \( ax + by = c \)
• Test point method: Substitute a point \( (x_0, y_0) \) not on the boundary line into the inequation to determine which side to shade
• Interval notation: \( (a, b), [a, b), (a, b], [a, b] \) depending on inequality
• Set-builder notation: \( { x \mid \text{condition on } x } \)