Notes On Graphical Solution of Linear Inequalities in Two Variables - CBSE Class 11 Maths

A linear inequality in two variables consists of two variables.

ax + by < c

The value of the variables for which the inequality holds true is called the solution of the inequality.


Ex: 2x + 3y < 8
Let x = 1

LHS: 2(1) + 3(1) = 2 + 3 =5

LHS: 5 < RHS: 8

x = 1 and y = 1 is a solution of the inequality since the statement of the inequality holds true.

Let x = 2, y = 1

LHS: 2(2) + 3(1) = 4 + 3 = 7

LHS: 7 < RHS:8


x = 2 and y = 1 is a solution of the inequality since the statement of the inequality holds true.

Let x = 1, y = 2

LHS: 2(1) + 3(2) = 2 + 6 = 8

LHS: 8 ≮ RHS: 8

x = 1 and y = 2 is not a solution of the inequality since the statement of the inequality holds true.

Graphical representation of a linear inequality in two variables

Consider the inequality 2x + 3y < 8



The line representing the equation 2x + 3y < 8 is drawn on the graph.


This line divides the Cartesian plane into two parts, the lower half plane and the upper half plane.

Plot the solutions of the inequality; also plot the point that is not the solution of the inequality.


The solutions of the given inequality lie on only one side of the line.

The region where the solutions are located is called as the solution region.

For the given inequality, the solution region is the lower half region.

A vertical line divides the plane into left and right half planes.

A non-vertical line will divide the plane into an upper half plane and a lower half plane.

Consider the inequality, ax + by > c.

Consider two cases, (i) b > 0 (ii) b < 0

(i) b > 0

Draw the line ax + by = c on the Cartesian plane.

Let (p, q) be a point on the line.

From the figure, we can observe that r is greater than q.

r > q

br > bq

ap + br > ap + bq

The statement is true for the point (p, r).

All the points lying in the half plane above the line ax + by = c satisfies the inequality ax + by > c,, provided b > 0.


In case of b < 0, the solution lies in the half plane below the line, ax + by = c.

Summary

A linear inequality in two variables consists of two variables.

ax + by < c

The value of the variables for which the inequality holds true is called the solution of the inequality.


Ex: 2x + 3y < 8
Let x = 1

LHS: 2(1) + 3(1) = 2 + 3 =5

LHS: 5 < RHS: 8

x = 1 and y = 1 is a solution of the inequality since the statement of the inequality holds true.

Let x = 2, y = 1

LHS: 2(2) + 3(1) = 4 + 3 = 7

LHS: 7 < RHS:8


x = 2 and y = 1 is a solution of the inequality since the statement of the inequality holds true.

Let x = 1, y = 2

LHS: 2(1) + 3(2) = 2 + 6 = 8

LHS: 8 ≮ RHS: 8

x = 1 and y = 2 is not a solution of the inequality since the statement of the inequality holds true.

Graphical representation of a linear inequality in two variables

Consider the inequality 2x + 3y < 8



The line representing the equation 2x + 3y < 8 is drawn on the graph.


This line divides the Cartesian plane into two parts, the lower half plane and the upper half plane.

Plot the solutions of the inequality; also plot the point that is not the solution of the inequality.


The solutions of the given inequality lie on only one side of the line.

The region where the solutions are located is called as the solution region.

For the given inequality, the solution region is the lower half region.

A vertical line divides the plane into left and right half planes.

A non-vertical line will divide the plane into an upper half plane and a lower half plane.

Consider the inequality, ax + by > c.

Consider two cases, (i) b > 0 (ii) b < 0

(i) b > 0

Draw the line ax + by = c on the Cartesian plane.

Let (p, q) be a point on the line.

From the figure, we can observe that r is greater than q.

r > q

br > bq

ap + br > ap + bq

The statement is true for the point (p, r).

All the points lying in the half plane above the line ax + by = c satisfies the inequality ax + by > c,, provided b > 0.


In case of b < 0, the solution lies in the half plane below the line, ax + by = c.

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