Linear Inequalities - Detailed Notes for JEE preparation

1. Types of Linear Inequalities:

- Linear inequalities with one variable:

An inequality involving a single variable, where either the greater than (>) or less than (<) symbol is used.

Examples:

• 2x + 5 > 11
• 3x - 4 < 10

- Linear inequalities with two variables:

An inequality involving two variables, with the region satisfying the inequality represented as a shaded region on a graph.

Examples:

• 3x + 4y ≤ 12
• x - 2y ≥ 6

- Linear inequalities with three variables:

An inequality involving three variables, where the region satisfying the inequality is represented as a solution set in three-dimensional space.

Example:

• 2x + y - 3z ≤ 15

2. Graphing Linear Inequalities:

- Plotting points on a number line:

• To graph a linear inequality with one variable, plot points on a number line based on the values that satisfy or do not satisfy the inequality.

Examples:

• For the inequality x > 3, plot the points 2, 3, and 4 on the number line. The point 3 will be an open circle, indicating it does not satisfy the inequality, and the points 2 and 4 will be filled circles, indicating they do satisfy the inequality.

- Graphing linear equations:

• To graph a linear equation with two variables, find the x- and y- intercepts, then draw a straight line through the points.

Example:

• To graph the equation y = 2x + 3, find the x-intercept by setting y = 0 and solving for x, and the y-intercept by setting x = 0 and solving for y. Then, plot these points and draw the line through them.

- Shading regions that satisfy the inequality:

• To shade the regions that satisfy a linear inequality with two or three variables, determine which side of the boundary line (linear equation) satisfies the inequality and shade the appropriate region accordingly.

Examples:

• For the inequality y ≤ 2x + 3, shade the region below the line y = 2x + 3.
• For the inequality 2x - y + z ≥ 5, shade the region that is on or above the plane 2x - y + z = 5.

3. Solving Linear Inequalities:

- Solving linear inequalities with one variable:

• To solve a linear inequality with one variable, isolate the variable on one side of the inequality by applying algebraic operations like addition, subtraction, multiplication, or division.

Example:

• To solve the inequality 2x + 5 > 11, subtract 5 from both sides, then divide by 2, to get x > 3.

- Solving linear inequalities with two variables:

• To solve a linear inequality with two variables, use graphical or algebraic methods to determine the region that satisfies the inequality.

Graphical method:

• Graph the boundary line (linear equation) and shade the appropriate region to show the solution.
• Example:
• To solve the inequality 3x + 4y ≤ 12, graph the line 3x + 4y = 12 and shade the region below the line.

(Algebraic method)

• Isolate one variable in terms of the other using algebraic operations, then specify the range of values for that variable that satisfies the inequality.
• Example:
• To solve the inequality 3x + 4y ≤ 12, solve for y to get 4y ≤ -3x + 12, or equivalently y ≤ (-3/4)x + 3. This specifies that y must be less than or equal to (-3/4)x + 3 for the inequality to hold.

- Solving linear inequalities with three variables using graphical method:

• Graph the boundary plane (linear equation) in three-dimensional space and determine the region that satisfies the inequality.

Example:

• To solve the inequality 2x + y - 3z ≤ 15, graph the plane 2x + y - 3z = 15 and determine the region that is on or below the plane.

- Solving linear inequalities with three variables using algebraic method:

• Express one variable in terms of the other two and substitute it into the inequality to obtain a two-variable inequality.
• Solve the resulting two-variable inequality graphically or algebraically.

Example:

• To solve the inequality 2x + y - 3z ≤ 15, solve for z in terms of x and y: 3z ≤ 2x + y - 15, or equivalently z ≤ (2/3)x + (1/3)y - 5. Then, substitute this expression for z into the original inequality to obtain 2x + y - 3((2/3)x + (1/3)y - 5) ≤ 15, which simplifies to x ≤ 5. Thus, the solution set is given by the region x ≤ 5 in the xy-plane.

4. Applications of Linear Inequalities:

Linear programming problems:

• Inequalities can be used to model and solve linear programming problems, which involve optimizing certain objectives while satisfying a set of constraints represented by linear inequalities.

Optimization problems:

• Use inequalities to find maximum or minimum values of functions subject to certain constraints, such as resource limitations or production capacities.

Break-even analysis:

• Inequalities are used to determine the break-even point in business scenarios, which is the point where revenue equals costs, by setting up and solving appropriate inequalities.

Profit maximization and cost minimization problems:

• Linear inequalities can be employed to model and solve problems related to maximizing profits or minimizing costs in various business and economic settings.

Inequalities in real-world situations:

• Inequalities have applications in various real-world scenarios, including investment allocation, budgeting, resource allocation, and other areas where optimizing decisions under constraints is required.

5. Properties and Theorems:

- Properties of inequalities:

• Properties of inequalities include the transitive property (if a < b and b < c, then a < c), multiplication and division properties (multiplying or dividing both sides of an inequality by the same positive constant preserves its direction), and addition and subtraction properties.

- Transitive property:

• If a < b and b < c, then a < c.

- Multiplication property:

• If a < b and c > 0, then ac < bc.
• If a < b and c < 0, then ac > bc.

- Division property:

• If a < b and c > 0, then a/c < b/c.
• If a < b and c < 0, then a/c > b/c.

- Addition property:

• If a < b and c < d, then a + c < b + d.

- Subtraction property:

• If a < b and c < d, then a - c < b - d.