Linear Inequalities

5.1 Introduction

In earlier classes, students have studied equations involving one or two variables and solved problems by translating statements into equations. However, not all real-world problems can be expressed as equations. Some statements involve comparisons rather than equalities, such as 'less than', 'greater than', 'less than or equal to', or 'greater than or equal to'. These comparisons are represented by inequalities using the symbols <, >, ≤, and ≥. For example, the height of all students in a class might be less than 160 cm, or a classroom might accommodate at most 60 tables or chairs. These statements cannot be represented by equations but by inequalities. This chapter introduces linear inequalities in one and two variables, which are fundamental in various fields such as science, mathematics, statistics, economics, and psychology. Understanding inequalities helps in solving practical problems where quantities are bounded or constrained but not necessarily equal.

• Equations involve equality; inequalities involve comparison.
• Inequalities use symbols <, >, ≤, ≥ to relate expressions.
• Real-world problems often require inequalities rather than equations.
• Linear inequalities involve expressions of degree one in variables.
• Inequalities are useful in diverse fields like economics and statistics.
• This chapter focuses on linear inequalities in one and two variables.
• 📌 Inequality: A mathematical statement relating two expressions using <, >, ≤, or ≥.
• 📌 Linear inequality: An inequality involving linear expressions (degree one) in variables.

5.2 Inequalities

This section introduces inequalities through practical examples and formal definitions. Consider Ravi who has ₹200 to buy rice packets priced at ₹30 each. If x is the number of packets, the total cost is 30x. Since he must buy whole packets, he may not spend the entire ₹200, leading to the inequality 30x < 200. This is not an equation because it does not involve equality but a strict inequality. Another example is Reshma who has ₹120 to buy registers and pens costing ₹40 and ₹20 respectively. If x and y are the numbers of registers and pens, the total cost is 40x + 20y ≤ 120. This inequality includes the possibility of spending exactly ₹120 or less. Inequalities can be strict (< or >) or slack (≤ or ≥). They can involve one or two variables. Numerical inequalities compare numbers directly, e.g., 3 < 5, while literal inequalities involve variables, e.g., x < 5. Double inequalities combine two inequalities, such as 3 ≤ x < 5. The section also classifies inequalities into linear inequalities in one variable (ax + b < 0), in two variables (ax + by ≤ c), and quadratic inequalities (ax² + bx + c > 0), though the chapter focuses on linear inequalities only.

• Inequalities express relations using <, >, ≤, ≥, not just equality.
• Examples include budget constraints in buying items.
• Inequalities can be strict (<, >) or slack (≤, ≥).
• Numerical inequalities involve numbers; literal inequalities involve variables.
• Double inequalities express two simultaneous inequalities.
• Linear inequalities in one or two variables are the focus; quadratic inequalities are excluded.
• 📌 Strict inequality: Inequality with < or >, excludes equality.
• 📌 Slack inequality: Inequality with ≤ or ≥, includes equality.
• 📌 Double inequality: Two inequalities combined, e.g., 3 ≤ x < 5.