Class 11 Maths Linear Inequalities Graphical Solution of Linear Inequalities in 2 variables

Graphical Solution of Linear Inequalities in 2 variables

A line divides the Cartesian plane into two equal parts. A point in Cartesian plane will either lie on a line or will lie in either of half planes. The region containing all the solutions of an inequality is called the solution region.

 

In order to identify the half plane represented by an inequality, it is just sufficient to take any point (a, b) (not online) and check whether it satisfies the inequality or not. If it satisfies, then the inequality represents the half plane and shade the region which contains the point, otherwise, the inequality represents that half plane which does not contain the point within it. For convenience, the point (0, 0) is preferred.

Example: x+2y >9

Steps for find solution region for a linear inequality in 2 variables

  1. Replace the inequality sign with equal sign & plot the graph. In this case plot graph for x+2y =9. Green line represent x+2y =9
  2. Take any point on the graph. Here we took (10, 10) & check if satisfies the linear inequality. In this case x+2y >9. If yes, then the region where this assumed point lies is the solution region.
  3. In case of Slack inequality (≥ or ≤) use solid line, since the points on the line is included in the solution set.
  4. In case of Strict inequality (> or <) use dotted line, since points on the line is not included in solution set.

 

In case of multiple linear inequalities, the region common to all the inequalities is the solution region.

Numerical: Solve the following system of inequalities graphically 5x + 4y ≤ 40    &   x > 2     &   y ≥ 3

Solution: Steps

  1. Draw lines for 5x + 4y = 40    &   x= 2     &   y = 3
  2. Find the solution region for each of these linear inequalities
  3. Find common region. Common region is the solution region.

 

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