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Linear inequalities in two variables

It is quite simple, that when there are two variables in a linear inequality, it is addressed as linear inequality in two variables.

Graphical Solution of Linear Inequalities in Two-Variables

The solution of inequalities could be represented as the graph of inequalities. We are already known to the fact that a line tends to divide the two-dimensional Cartesian plane into two parts called halve-planes. As shown in the figure below, a vertical line divides the plane into the right and left halves.

inequalities (graph).jpg

Also, a non-vertical line divides the Cartesian plane into lower and upper half as shown in the figure below:

non vertical division.jpg

Any point on the Cartesian plane is supposed to either:

  • Lie in the line
  • Lie within the half plane – I; or
  • Lie within the half plane – II

In order to graph any linear inequality in two variables (consider and ), you would be required to first bring to one side. Further, consider a related equation that may be obtained by changing the prevailing inequality sign into equal sign. The graph of this equation is likely to be a line.

When the inequality is strict (< or >), the graph for it would be a dashed line. Also, when the inequality is not strict (≤ and ≥), the graph would depict a solid line.

Now, pick up one point that is not on the line; ((0,0) is generally the easiest); and make out whether or not these coordinates would satisfy the inequality. If it does, then shade the half plane consisting of the point. If it doesn’t, then shade the other half.

Consider graphing an inequality

The line tends to be already in the slope-intercept form, with y on the left side alone. Its slope is 4 and the y intercept for it is -2. Graphing it is therefore straight forward. Here, since we have a “less than or equal to” inequality, we would make a solid line.

graphing inequalities.gif

Now, consider substituting x=0, y=0; in order to decide whether (0, 0) could satisfy the inequality.

0 ≤ 4(0) – 2

0 ≤ – 2

half-plane.gif

Since this would be false, you will have to shade the half-plane which does not include the coordinates (0, 0).

With the help of above-explained theories, you may carry on with solutions and graphical representations of linear inequalities in two variables correctly and come up with the most appropriate solutions to these problems.

 

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