Linear Inequalities — Study Notes

1. Introduction

Inequalities express the idea that one quantity is greater than or less than another. While equations state that two expressions are equal, inequalities allow for a range of values to satisfy a condition. Linear inequalities in one or two variables are used to model constraints in real life, such as budgeting, production limits, and allocation problems.

This chapter defines inequalities involving linear expressions, explains the rules for solving them, and presents their solution sets in both algebraic and graphical forms. In one variable, solutions are intervals on the number line. In two variables, the solution set is a half-plane bounded by a straight line, which forms the foundation of linear programming studied in higher classes.

Linear inequalities appear everywhere: age limits, speed limits, profit margins, and resource constraints can all be expressed as inequalities. The graphical method developed here, where a region satisfying multiple inequalities is identified, is the basis for optimisation in economics and operations research. Mastering the rules for reversing inequality signs when multiplying or dividing by a negative number is the key skill of this chapter.

2. Inequalities: Basic Definitions

A statement involving the symbols < (less than), <= (less than or equal to), > (greater than), and >= (greater than or equal to) is called an inequality.

For example, x < 4, 3x + 2 > 8, and 2x - 1 <= 5 are all inequalities. An inequality containing only one variable and where the variable has degree one is called a linear inequality in one variable.

Two real numbers a and b are compared as follows: a > b means a - b is positive, and a < b means a - b is negative. The rules of arithmetic of inequalities rest on these sign properties of differences.

3. Rules for Solving Inequalities

The following rules are used to transform an inequality into a simpler equivalent one:

• Rule 1: Adding the same number to both sides does not change the inequality. If a > b, then a + c > b + c.
• Rule 2: Multiplying both sides by a positive number preserves the inequality. If a > b and c > 0, then ac > bc.
• Rule 3: Multiplying both sides by a negative number reverses the inequality. If a > b and c < 0, then ac < bc.

The same rules apply for <= and >=. The third rule is the most error-prone: whenever we multiply or divide an inequality by a negative number, the direction of the inequality symbol must be reversed.

4. Solving Linear Inequalities in One Variable

To solve an inequality, apply the above rules to isolate the variable. The solution set is written as an interval on the number line.

Example: Solve 5x - 3 < 3x + 1. Bring variable terms together: 5x - 3x < 1 + 3, giving 2x < 4, so x < 2. The solution set is (-infinity, 2).

Example: Solve -2x > 6. Dividing both sides by -2 (a negative number) reverses the inequality: x < -3. The solution set is (-infinity, -3).

Interval Notation

The solution sets are written using intervals:

• x >= a: [a, infinity)
• x > a: (a, infinity)
• x <= a: (-infinity, a]
• x < a: (-infinity, a)
• a <= x <= b: [a, b]
• a < x < b: (a, b)

5. System of Linear Inequalities in One Variable

To solve a system of linear inequalities, solve each inequality separately and take the intersection of the solution sets. Graph each solution on the number line and find the common region.

Example: Solve x + 3 > 1 and 2x - 4 < 2. First inequality: x > -2. Second inequality: 2x < 6, so x < 3. The common solution is -2 < x < 3, i.e., the interval (-2, 3).

If the intersection of the solution sets is empty, the system has no solution.

6. Linear Inequalities in Two Variables

An inequality of the form ax + by + c < 0, ax + by + c > 0, ax + by + c <= 0, or ax + by + c >= 0, where a and b are not both zero, is a linear inequality in two variables.

The solution set is a region of the coordinate plane. The line ax + by + c = 0 divides the plane into two half-planes. To decide which half-plane is the solution, test a point not on the line, such as (0, 0) when it is not on the line.

Steps to Graph ax + by + c > 0

1. Draw the line ax + by + c = 0 as a dotted line (since points on the line are not included when the inequality is strict).
2. Choose a test point not on the line, e.g., (0, 0).
3. Substitute into the inequality. If true, shade the half-plane containing the test point; otherwise shade the other half-plane.

For <= or >=, draw the line as a solid line since boundary points are included.

7. System of Linear Inequalities in Two Variables

When more than one inequality must be satisfied simultaneously, graph each inequality and identify the region common to all solution sets. This common region is called the feasible region.

Example: Solve x + y < 6 and x >= 1, y >= 1. Draw the line x + y = 6 as a dotted line, test (0, 0): 0 < 6 is true, so shade below the line. Then shade the region where x >= 1 (right of x = 1) and y >= 1 (above y = 1). The feasible region is the intersection of these three half-planes.

Important Cases

• If the feasible region is empty, the system has no solution.
• If the constraints produce an unbounded region, the solution set is unbounded.
• Corner (vertex) points of the feasible region are significant in linear programming because optima occur there.

8. Applications of Linear Inequalities

Word problems translate real situations into inequalities. The key is to identify the unknown, form the inequality from the given condition, and solve it.

Example: A man earns at least 20000 rupees a month. If x is his monthly income, then x >= 20000. Example: In a test, a student needs at least 40 marks to pass. If m is the marks, m >= 40.

In two-variable problems, such as determining the maximum profit subject to resource constraints, the inequalities define the feasible region and the objective is optimised over it. These ideas are formalised in linear programming in Class 12.

Quick Revision Tables

Table 1: Rules of Inequality and Their Effects

Table 2: Interval Notation

Mind Map

Important Diagrams (SVG)

Diagram 1: Number Line Solutions of Inequalities in One Variable

Diagram 2: Half-Plane Solution of a Linear Inequality in Two Variables

Common Mistakes

1. Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. For example, -2x > 6 gives x < -3, not x > -3.
2. Adding or subtracting incorrectly while isolating the variable, especially when bringing terms across the inequality sign.
3. Using an open circle when the endpoint is included. If the inequality is <= or >=, the endpoint circle must be filled.
4. Shading the wrong half-plane. Always test a point such as (0, 0) before deciding which side of the line to shade.
5. Drawing the boundary line solid when the inequality is strict. Use a dotted line for < and >, a solid line for <= and >=.
6. Taking the union instead of the intersection of solution sets for a system of inequalities. A system requires all inequalities to hold simultaneously.
7. Mistaking the direction: x > a and a < x are the same inequality. Reading the inequality backwards must not change the solution set.
8. Forgetting to exclude points where a rational expression is undefined when combining inequalities with fractions.

Exam Tips

1. Always reverse the sign when multiplying or dividing an inequality by a negative number, and state this step explicitly in your answer.
2. To write the solution set, convert to interval notation with the correct brackets: round for strict, square for non-strict.
3. For systems in one variable, solve each inequality and draw both number lines to find the common interval visually.
4. In two-variable graphing, choose (0, 0) as the test point whenever it is not on the boundary line; it makes the arithmetic simplest.
5. For word problems, first identify the unknown and then translate phrases like "at least", "at most", "more than", and "not exceeding" into the correct symbol (>=, <=, >, <).
6. When asked to find the solution of a system with no common region, clearly state that there is no solution.
7. Practise problems involving both x >= 0 and y >= 0 constraints, since these appear in most application questions and define the first quadrant region.

Conclusion

Linear inequalities generalise the concept of equations to comparisons and constraints. The ability to solve inequalities in one variable, express solutions as intervals, and graph solution regions in two variables is a fundamental skill for mathematics and its applications. The half-plane method introduced here directly leads to linear programming, where optimal solutions are sought over feasible regions defined by systems of inequalities. Paying careful attention to the reversal rule and the test-point method will make this chapter straightforward and will build skills that are heavily used in economics, optimisation, and higher mathematics.