Linear Inequalities in Two Variables

Any two real numbers or algebraic expressions joined by the symbols “<”, “>”, “≥”, “≤”, form an inequality. The symbols “≥” and “≤” stand for greater than or equal to and less than or equal to in inequalities. A number line is a practical and visually appealing way to represent the solutions to linear inequalities in a single variable.

A statement that one expression is greater than or less than another is an inequality. A linear inequality is an expression that uses an inequality symbol to compare two values. The symbols which are used to represent inequality are as follows:

• Not equivalent (≠)
• Less than (<)
• Greater than (>)
• Less than or equal to (≤)
• Greater than or equal to (≥)
   

The solution is the set of ordered pairs that satisfy the inequality. The required solutions will be an ordered pair (x, y) that satisfies the statement if ax+by>c is a linear inequality with two variables, x and y. Solving linear inequalities in two variables is the same procedure as solving linear equations, except that the solution is a region, not a single point. But instead of getting one answer, you will find a whole area of points that satisfy the inequality. For example, we can check the solution if a point is in a solution to the inequality 2x+4y>3 by entering the values of x and y.
Let x = 1 and y = 2.
Using LHS, we have
2(1) + 4(2) = 2 + 8 = 10
The ordered pair (1, 2) satisfies the inequality 2x+4y>3
Since, 10 > 3, therefore, it is satisfied.
 

First inequality

y ≤ -x + 4 
Step 1: Boundary line conversion to equation:
y=-x+4
Step 2: We draw a solid line to show that the points are on the line, are part of the solution (including the boundary) because it is ≤.
Step 3: Point of test (0, 0) replace the inequality with 0 ≤ -0+4=0 ≤ 4 (real)
Therefore, we shade the area below the line that contains (0, 0).

Second inequality

y>2x-3
Step 1: Boundary line conversion to equation:
y=2x-3
Step 2: Draw a dashed line (not including the boundary) because it is >.
Step 3: Point of test (0, 0) replace the inequality with
0 > 2 (0) - 3 = 0 > -3 (real)
Therefore, we shade the area above the line that contains (0, 0).
 

Statements containing two distinct variables are known as linear inequalities in two variables. The symbols such as “<“ (less than), “>” (greater than), “≤” (less than or equal to), “≥” (greater than or equal to). Let’s look at a graphical example of how to solve such an expression.
The two examples of linear inequalities shown in the image are listed below. The following are the graphs for y>2x-3 and y ≤ -x + 4: 

Key Formula for Linear Inequalities in Two Variables

The linear inequalities of two algebraic expressions compare when they are not equal, using inequality symbols like, <, >, ≥, or ≤
The standard formula is as follows:

• ax+by<c
• ax+by ≤ c
• ax+by>c
• ax+by ≥ c

Where x and y are variables and a, b, and c are real numbers.
 

Linear inequalities in two variables help us make decisions within limits. Let us see how it works

Cost restrictions and budgeting

The x representing the cost of fruits and y representing the cost of vegetables, a person can spend no more than ₹1000 on fruits and vegetables. This can be written as x+y ≤ 1000 is the inequality.

Planning for business profits

To meet a goal, a business needs to sell at least 50 units of product A (x) and at least 30 units of product B (y). The inequality is now written as x ≥ 50 and y ≤ 30.

Production capacity of the factory

If a chair takes two hours (x) to produce and a table takes five hours (y), and the factory can only work 100 hours, the inequality is 2x+5y ≤ 100.

Planning for event capacity

If a hall can accommodate 300 people and there are x adults and y children attending an event, the inequality is x + y ≤ 300.

Allocating study time

A student studies for x hours in math and y hours in science, for a total of no more than 6 study hours per day. Which can be represented by the inequality  x+y ≤ 6.