Linear Inequalities In Two Variables

Difference between equation and inequality

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), ‘\(\leq\)’(less than or equal to), ‘\(\geq\)’ (greater than or equal to) are called inequalities.

Inequalities - Linear Inequalities In Two Variables

Following example validates the difference between equation and inequality:

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

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,

\( \geq \) 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 for double inequality.

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

Example: \( 2x + 3 \leq 6, 2x + 3y \geq  6\)

Slack inequality: Mathematical expressions involve only \(’\leq'\) or \( \geq\) are called slack inequalities.

Example: \( 2x + 3 \leq 6, 2x + 3y \geq 6 \)

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, \( 2x + 3y \geq 6  \) is a linear inequality in two variables because there are two variables ‘x’ and ‘y’ are present in the expression.

Note: \( 4x^2 + 2x + 5 < 0 \) 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.

Example:  Classify the following expressions into

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

\( 5x < 6, 8x + 3y \leq 5, 2x – 5 < 9 , 2x \leq 9 , 2x + 3y < 10 \)

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

Now learn concept in depth along with NCERT Solution for Linear Inequalities.


Practise This Question

Which of the following shows 114 and 258 on a number line?