Linear Inequalities In Two Variables

Linear inequalities in two variables represent the inequalities between two algebraic expressions where two distinct variables are included. Basically, in linear inequalities, we use greater than (>), less than (<), greater than or equal (≥) and less than or equal (≤) symbols, instead of using equal to a symbol (=). For linear equalities, we use ‘Equal to’ symbol or show equalities between any two expression or numbers. For example, x+3=y, x=2-y, etc.

Linear Inequalities Definition: Any two real numbers or two algebraic expressions associated by the symbol ‘<’, ‘>’, ‘≤’ or ‘≥’ form an inequality. For example, 9<11, 18>17 are examples of numerical inequalities. and x+7>y, y<10-x, x ≥ y > 11 are the examples of algebraic or literal inequalities.

The symbols ‘<‘ and ‘>’ represent the strict inequalities and the symbols ‘≤’ and ‘≥’ represent slack inequalities. To represent linear inequalities in one variable in a number line is a visual representation and is a convenient way to represent the solutions of the inequality. Now, we will discuss the graph of a linear inequality in two variables.

Also, read:

Graphical Solution of Linear Inequalities in Two Variables

An equation is a mathematical expression which involves “=” symbol. The right-hand side of the expression is equal to the left-hand side of the expression.

The statements involving symbols like ‘<’(less than), ‘>’ (greater than), ‘≤’’(less than or equal to), ‘≥’ (greater than or equal to) are called inequalities.

Linear Inequalities In Two Variables

Example:

Following example validates the difference between equation and inequality:

Statement 1: The distance between your house and school is exactly 4.5 kilometres,

The mathematical expression of the above statement is,

x = 4.5 km, where ‘x’ is the distance between house and the school.

Statement 2: The distance between your house and the school is at least 4.5 kilometers.

Here, the distance can be 4.5 km or more than that. Therefore the mathematical expression for the above statement is,

x ≥ 4.5 km, where ‘x’ is a variable which is equal to the distance between house and the school.

Types of inequalities

Numerical inequalities: If only numbers are involved in the expression, then it is a numerical inequality.

Example: 10 > 8, 5 < 7

Literal inequalities: x < 2, y > 5, z < 10 are the examples for literal inequalities.

Double inequalities: 5 < 7 < 9 read as 7 less than 9 and greater than 5 is an example of double inequality.

Strict inequality: Mathematical expressions involve only ‘<‘ or ‘>’  are called strict inequalities.

Example: 2x + 3 < 6, 2x + 3y > 6

Slack inequality: Mathematical expressions involve only ‘≤′ or ‘≥’ are called slack inequalities.

Example: 2x + 3 ≤ 6, 2x + 3y ≥ 6

In the above examples,  2x + 3 < 6 is a linear inequality in one variable because ‘x’ is the only one variable present in the expression.

Similarly, 23y ≥ 6  is a linear inequality in two variables because there are two variables ‘x’ and ‘y’ are present in the expression.

Note: 4x2 + 2x + 5 < 0 is not an example of linear inequality in one variable, because the exponent of x is 2 in the first term. It is a quadratic inequality.

Examples

1.) Classify the following expressions into:

  • Linear inequality in one variable.
  • Linear inequality in two variables.
  • Slack inequality.

5x < 6, 8x + 3y ≤ 5, 2x – 5 < 9 , 2x ≤ 9 , 2x + 3y < 10.

Solution:

Linear inequality in one variable Linear inequality in two variables Slack inequality
5x < 6 8x + 3y ≥ 5 8x + 3y ≥ 5
2x – 5 < 9 2x + 3y < 10 2x ≤ 9
2x ≤ 9

 

2.) Solve y < 2 graphically.

Solution: Graph of y = 2. So we can show it graphically as given below:

Graphical Solution of Linear Inequalities in Two Variables

Let us select a point, (0, 0) in the lower half-plane I and putting y = 0 in the given inequality, we see that:
1 × 0 < 2 or 0 < 2 which is true.
Thus, the solution region is the shaded region below the line y = 2.

Hence, every point below the line (excluding all the points on the line) determines the solution of the given inequality.

Linear Inequalities in Two Variables Word Problem

In an experiment, a solution of hydrochloric acid is to be kept between 25° and 30° Celsius. What is the range of temperature in degree Fahrenheit if conversion formula is given by C = 5/9 (F – 32), where C and F represent the temperature in degree Celsius and degree Fahrenheit, respectively.

Solution: As per the question it is given:

25<C<30

Now if we put C = 5/9 (F – 32), we get;

25 < 5/9 (F – 32) < 30

or

9/5 x 25 < F – 32 < 30 x 9/5

45 < F -32< 54

77 < F < 86

Thus, the required range of temperature is between 77° F and 86° F

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