Ex 14.1 from Chapter 10 of Class 10 Mathematics is an introductory chapter on linear inequalities. A linear inequality is essentially a mathematical expression that comprises variables as well as numerical constants and involves the less-than (), less-than-or-equal-to (), greater-than (), or greater-than-or-equal-to () signs. These inequalities plays a critical role in various mathematical applications, including optimization problems, and are foundational to any student pursuing higher education in mathematics, engineering, economics, computer science, etc.

NCERT Solutions for Class 10 Maths Chapter 2 Exercise 2.2 - Access PDF ...
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Understanding Linear Inequalities

Ex 14.1 Class 10 By Direct Method: Simplifying Linear Inequalities

Solving linear inequalities involves finding solutions for the unknown variables (usually denoted by x or y) that satisfy the given inequality. The solution process entails isolating the variable on one side of the inequality while accounting for any relevant operations conducted on both sides. This is similar to the process used for solving equations. However, it is worth noting that unlike equations, which have a single unique solution, inequalities typically have infinitely many solutions that form a range of values.

Types of Linear Inequalities

One-variable linear inequalities are characterized by having only one variable and involve any of the four inequality operators (<, ≤, >, ≥).

Two-variable linear inequalities incorporate two variables within their equations and are typically represented graphically on a coordinate plane.

Graphical Representation

Graphically representing a linear inequality requires:

  1. Graphing the corresponding equation (without the inequality sign) as a line.

  2. Determining if the shaded region is above or below the graphed line based on the inequality sign.

  • Above the line if (>) or ≥) used

  • Below the line if (<) or ≤) used

NCERT Solutions for class 10 Maths Chapter 2 Exercise 2.2
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Solving Two-Variable Linear Inequalities

Solving two-variable linear inequalities graphically involves:

  1. Graphing the boundary line as a dashed line.

  2. Shading the appropriate region based on the inequality sign used.

  3. Points inside the shaded region are considered solutions to the inequality.

Steps for Solving Linear Inequalities

  1. Isolate the variable

  2. Perform only legal operations on both sides to ensure equivalence.

  3. Express the solution in terms of the lone variable.

Applications of Linear Inequalities

Linear inequalities have diverse applications across numerous domains, including:

  1. Business Optimization: Minimizing production costs or maximizing profits by determining variable values satisfying given constraints.

  2. Decision-Making: Defining boundary conditions or thresholds for choices based on specific criteria

  3. Physics Simulations: Establishing the limits of variables in equations describing physical behaviors or processes.

Ex 14.1 Class 10 By Direct Method

Conclusion

Linear inequalities are an indispensable part of the mathematics curriculum. They extend the understanding of equations to explore concepts like ranges of values and graphical representations. This article introduced the basics of ex 14.1 from Class 10 Mathematics, providing guidance on solving linear inequalities. Through logical and well-explained steps, readers can enhance their problem-solving skills and gain a deeper understanding of this foundational mathematical concept.


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