Linear Inequalities

Notes on Linear Inequalities

1. Introduction to Linear Inequalities

• Linear inequalities are mathematical expressions where two values or expressions are compared using inequality symbols: <, >, ≤, or ≥.
• Unlike equations, inequalities express a range of possible values rather than a single solution.
• For instance, statements like "all students' height is less than 160 cm" contain inequalities because they describe conditions without strict equality.

2. Definitions and Examples

• An inequality can be a simple numerical comparison (3 < 5) or involve variables, such as ax + b < c.
• Types of inequalities include:
  • Strict inequalities: (e.g., x < 5, y > 2)
  • Non-strict inequalities: (e.g., x ≤ 4, y ≥ 2)
  • Double inequalities: e.g., 3 < x < 5, which indicates x lies between 3 and 5.
• A subset of inequalities are linear inequalities, which include coefficients of the variable terms that are constants.

3. Algebraic Solutions to Linear Inequalities

• To solve linear inequalities, we employ similar methods as linear equations with the following rules:
  • Rule 1: You can add or subtract the same value from both sides of an inequality without changing its direction.
  • Rule 2: You can multiply or divide both sides by a positive number without changing the direction. However, if you multiply or divide by a negative number, the direction of the inequality reverses.

Examples of Solving Inequalities:

Example 1: Solve 30x < 200.

• Dividing both sides by 30: x < 200/30, hence x < 20/3.
• If x is natural numbers: the solution set is {1,2,3,4,5,6}.
• If x is an integer: solution set includes {..., -3, -2, -1, 0, 1, 2, 3, 4, 5, 6}.

Example 2: Solve the inequality 5x – 3 < 3x +1.

• Rearranging gives: 2x < 4, hence x < 2.
• Solution varies if considering integers or reals, e.g., integers: {..., -4, -3,..., 0, 1} and reals: (-∞, 2).

4. Graphical Representation

• To visually represent inequalities on a number line:
  • Use a open circle for < or >, indicating that the point is not included.
  • Use a closed circle for ≤ or ≥, indicating inclusion of the point.
  • For example, for x < a, you depict it as a line extending left from a with an open circle, while x ≤ a has a closed circle on a with a line extending left.

5. System of Inequalities

• Sometimes, you may deal with multiple inequalities simultaneously. For example, to solve -8 ≤ 5x - 3 < 7, solve each part separately:
  • -8 ≤ 5x - 3 results in: -5 ≤ 5x → x ≥ -1
  • 5x - 3 < 7 results in: 5x < 10 → x < 2.
  • The combined solution is -1 ≤ x < 2.

6. Word Problems and Applications

• Inequalities apply widely in real life – from economics to scientific experiments.
  • For instance, to maintain certain averages or proportions (e.g., a student’s average marks, comparative costs, etc.).
  • E.g., 62 + 48 + x ≥ 180 helps determine minimal required grades.

7. Exercises and Practice

• It is important to solve a variety of problems to gain proficiency in recognizing and solving inequalities. Exercises encompass finding the ranges of values that satisfy a given condition, illustrating their importance in various contexts.

8. Concluding Summary

• A pivotal part of mathematics, understanding linear inequalities sharpens analytical skills, critical for fields requiring quantitative reasoning and decision-making.

Key Points to Remember:

1. Inequality refers to expressions involving <, >, ≤, or ≥.
2. Strict inequalities do not include endpoint values, whereas non-strict inequalities do.
3. Rules of manipulation for inequalities differ when multiplying/dividing by negative numbers.
4. Solution sets consist of all values that satisfy the inequality, represented graphically.
5. Solve inequalities in appropriate number sets (naturals, integers, reals) based on context.
6. Inequalities can represent real-life problems, aiding in decision-making and planning.
7. Practice with worksheets/exercises solidifies understanding and application of concepts.

1. Inequality symbols include <, >, ≤, and ≥.
2. Strict inequalities don’t include endpoints; non-strict do.
3. Adding/subtracting equal amounts does not change the inequality.
4. Multiplying/dividing by a negative reverses the inequality sign.
5. Solution set represents all values satisfying the inequality.
6. Represent solutions graphically on number lines.
7. Inequalities apply to various real-life contexts.
8. Solve corresponding word problems to apply learned concepts.