In Class 11, linear inequalities are introduced as a part of the study of algebra and are an extension of the concept of linear equations. Here’s an overview:
1. **Linear Inequalities**: A linear inequality is an inequality that involves a linear expression in one or more variables. The standard form of a linear inequality in one variable is \(ax + b < 0\) or \(ax + b > 0\), where \(a\) and \(b\) are constants and \(x\) is the variable. In two variables, a linear inequality looks like \(ax + by < c\) or \(ax + by > c\), where \(a\), \(b\), and \(c\) are constants.
2. **Graphical Representation**: Linear inequalities can be represented graphically on the coordinate plane. The solution set of a linear inequality in two variables forms a region in the plane bounded by a line (the boundary) and shaded to indicate the side of the line that satisfies the inequality.
3. **Solving Linear Inequalities**: Solving linear inequalities involves finding the values of the variables that satisfy the inequality. The solution set is often expressed in interval notation or set notation.
4. **Properties of Inequalities**: The properties of inequalities, such as the addition and multiplication properties, are used to manipulate and solve linear inequalities. These properties are similar to those of equations but with some differences.
5. **Systems of Linear Inequalities**: A system of linear inequalities consists of two or more linear inequalities with the same variables. The solution to a system of linear inequalities is the intersection of the solution sets of the individual inequalities.
Understanding linear inequalities is important in various fields, including economics, optimization problems, and operations research, where constraints are often represented by inequalities.
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