RS Aggarwal Class 11 Solutions Chapter 7 Linear Inequations (In two variables)

A linear equation in two variables, like ax + by = c, represents a straight line on a graph where every point on the line is a solution. In contrast, a linear inequation, such as ax + by > c or ax + by ≤ c, represents an entire region or plane on one side of that line. The solution set is not just the line itself but an infinite number of points in the shaded area.

To graphically represent a linear inequation, follow these steps:

• First, treat the inequation as an equation to draw the boundary line.

• Draw a solid line if the inequation includes equality (≤ or ≥) or a dashed line for strict inequality (< or >).

• Choose a test point (like the origin (0,0), if it's not on the line) and substitute it into the original inequation.

• If the test point satisfies the inequation, shade the entire region containing that point. If not, shade the region on the opposite side of the line.

The type of line indicates whether the points on the line itself are part of the solution. A solid line is used for inequations with 'or equal to' (≤ or ≥), signifying that the points on the line are included in the solution set. A dashed line is used for strict inequations (< or >) to show that the points on the line are not part of the solution; the line only serves as a boundary for the solution region.

To solve a system of linear inequations, you must find the region that satisfies all inequations simultaneously. The steps are:

• Graph each linear inequation on the same coordinate plane, following the standard procedure for each.

• Carefully shade the solution region for each individual inequation.

• The final solution is the common shaded area or the region where all the individual shaded areas overlap. This overlapping region is known as the feasible region.

While the NCERT textbook provides a strong foundation for Linear Inequations, RS Aggarwal offers a more extensive range of problems with varying levels of difficulty. It includes more complex systems of inequations and word problems that require a deeper application of concepts. Solving these helps students master the graphical method, handle different types of constraints, and prepare thoroughly for school exams and competitive tests.

The feasible region is the graphical solution to a system of linear inequations. It is the overlapping area that contains all the points (x, y) that satisfy every single inequation in the system at the same time. This concept is crucial because it represents all the possible and valid solutions to a problem, especially in real-world applications like resource allocation or profit maximisation, which form the basis of Linear Programming.

A very common mistake is automatically using the origin (0,0) as the test point without checking if the boundary line passes through it. If the line for an equation like y = 2x or x - 3y = 0 passes through the origin, substituting (0,0) will result in 0 = 0, which gives no information about which side to shade. In such cases, you must choose a different, unambiguous test point, such as (1,0) or (0,1), to correctly identify the solution region.

Yes, absolutely. Linear inequations are widely used to model real-world scenarios involving constraints. For example, a student might have a time constraint for studying two subjects, expressed as x + y ≤ 5 hours. A business might have a budget constraint for manufacturing two different products, such as 100x + 200y ≤ 50000. The feasible region in such cases shows all the viable combinations that meet the given conditions.