Linear Inequations with two Variables

Objectives

At the end of this lesson, you should be able to:

Find the solution of a linear inequation with the formula $a x + b y < c$ .
Graph the region in the plane that represents the solution of any linear inequation of any two variables.

Introduction

An inequation with two variables is an inequation that can be written like:

$a x + b y < c$

or any expression of the previous form where, in the place of the symbol < includes any other symbol of inequality: > , ≤ o

where a, b and c are constants and x and y are variables. Solve and inequation in two variables consisting of finding all the pairs of values of (x,y) for those that equal the difference.

Like we saw in the tutorial of linear equations of a dimension, we can change the sign of the inequality by the same sign, we find an equation that comes to be the beginning of the solution of the inequality. For example, consider the following inequality: $y < x$ .

Changing the sign < for the sign = we find the equation: $y = x$ . The graph of this equation is a line that divides the plane in two regions as shown in the following figure:

We take any point in the red region, for example the point (1,-1). Where x=1 and y=-1. Such as -1<1, this pair of values satisfies the inequality: $y < x$ .

We take another point in the red region, for example the point (2,1). Where x=2 and y=1. Such as 1<2, this pair of values satisfies the inequality: $y < x$ .

Try to find a point in the red region where $y > x$ . You will see that it is not possible to find a point that fills these requirements.

In conclusion, any point of the red region that satisfies the inequality $y < x$ .

In the same way, any point in the yellow region, satisfies the inequality $y > x$ . Check it selecting points across the yellow region.

In general, at the change of the sign of inequality for the sign = we find an equation of a line that comes to be the beginning of the solution of the inequality.

Press the button below to practice visualizing how similarities divide the plane xy

General method for solving linear inequations with two variables

To solve an inequation of the form:

$a x + b y < c$

or any expression of the previous form that, in place of the symbol < includes any other symbol of inequality: > , ≤ or ≥, we continue with the following steps:

Replace the sign of inequality for the sign = $a x + b y = c$ and divide the cartesian plane taking the beginning of the line that represents the equation.
Take points of test in every region and verify if they satisfy the inequality.
Graph the solution, having the total that is the inequality ≥ or ≤ the beginning is included in the solution, in case the opposite of the beginning is not included.

Examples

Example 1:

Solve the following inequation $x + y < 4$

Solution:

Step 1: Replace the sign of inequality for the sign =, we get the following equation

x + y = 4

. To graph a line, it is sufficient to find the two points. One simple way of graphing the line is to find the intercepts with the axis:
To find the intercept with the axis x, we do y=0,

$\begin{matrix} x + y = 4 \\ x + 0 = 4 \\ x = 4 \end{matrix}$

To find the intercept with the axis y, we do x=0,

$\begin{matrix} x + y = 4 \\ 0 + y = 4 \\ y = 4 \end{matrix}$

The graph is the following. This line divides the plane in two regions R1 and R2.

Step 2: Take points of test in every region and verify is they satisfy the inequality.

Point of test in R1 (0,0)

$\begin{matrix} x + y < 4 \\ 0 + 0 < 4 \\ 0 < 4 \end{matrix}$

Like the expression is true, then this is the region that represents the solution of the inequation.

Like we already determined in the solution, it is not necessary to select a point of test in the other region. You can try any point in the other region that does not satisfy the inequality.

Step 3: Graph the solution. Like the sign of the inequality is < one cannot include the beginning as part of the solution. To denote this graphically, we utilize discontinuous lines in the beginning.

Example 2:

Solve the following inequation $2 x + 3 y \geq 6$

Solution:

Step 1: Replace the sign of inequality for the sign =, we find the following equation

2 x + 3 y = 6

. To graph a line, it is sufficient to find two points. A simple form of graphing the line is to find the intercepts with the axis:
To find the intercept with the axis x, we do y=0,

$\begin{matrix} 2 x + 3 y = 6 \\ 2 x + 3 (0) = 6 \\ 2 x = 6 \\ x = 3 \end{matrix}$

To find the intercept of the axis y, we do x=0,

$\begin{matrix} 2 x + 3 y = 6 \\ 2 (0) + 3 y = 6 \\ y = 2 \end{matrix}$

The graph of the line is the following. This line divides the plane in two regions R1 and R2.

Step 2: Take points of test in every region and verify if they satisfy the inequality.

Point of test in R1 (1,1)

$\begin{matrix} 2 x + 3 y \geq 6 \\ 2 (1) + 3 (1) \geq 6 \\ 5 \geq 6 \end{matrix}$

Like the expression is false, then this region is not the solution of the inequality.

Point of test in R2 (3,4)

$\begin{matrix} 2 x + 3 y \geq 6 \\ 2 (3) + 3 (4) \geq 6 \\ 18 \geq 6 \end{matrix}$

Like the expression is true, then this region is the solution of the inequality.

Step 3: Graph the solution. Like the sign of the inequality is ≥ one can include in the beginning as part of the solution. To denote this graphically, we utilize a continuous line in the beginning.

Example 3:

Solve the following inequality in two dimensions $x > 2$

Solution:

Step 1: Replace the sign of inequality for the sign =, we find the following equation

x = 2

. This equation corresponds to a vertical line with the intercept in the point x=3.

Step 2: Take points of test in every region and verify if it satisfies the inequality.

Point of test in R1 (0,0)

$\begin{matrix} x > 2 \\ 0 > 2 \end{matrix}$

Like the expression is false, then this region is not the solution of the inequality.

Point of test in R2 (3,3)

$\begin{matrix} x > 2 \\ 3 > 2 \end{matrix}$

Like the expression is true, then this region is the solution of the inequality.

Step 3: Graph the solution. Like the sign of the inequality is > it cannot be included in the beginning like part of the solution.

Press the button below to practice graphing regions of inequality in the plane xy.

Press the button below to practice exercises of regions of inequations in the plane xy.

Summary

Now that you ahve completed this lesson, you should be able to:

Find the solution of a linear inequality in the form of $a x + b y < c$ .
Graph the region in the plane that represents the solution of any linear inequality with two variables.