The linear equations and inequalities in one variable are important mathematical expressions and hold significant importance in solving problem statements for competitive exams. There is only a minute difference between solving inequalities and linear equations, but otherwise, both are the same. The linear equation does not change by dividing or multiplying both sides of the equation by the same negative integer. However, for linear inequalities, multiplying or dividing both sides of an inequality by the same negative integer reverses the inequality. There are graphical methods used for solving inequalities in one variable for finding linear equation solutions. When two expressions are related with less than (<), greater than (>), or less than equal to (≤) or greater than equal to (≥) sign, it is called an inequality. The linear function that involves inequality expression is called linear inequality. 

Linear inequalities in one variable: understanding linear equation, basic concepts

 The linear inequalities require an understanding of some important formulae. In general, the linear equation exists in the form – for example, 8a + 2 = 18, where 8 and 2 are integers and x is a variable, indicating the presence of only one variable. Hence, for solving inequalities in one variable, only one variable of a maximum of one order within a linear equation is to be solved.

Here are some of the examples of linear inequalities in one variable

  • 6x + 4 = 24 where x = 10/3 (it is a linear equation with a single variable x)
  • 5x + 6 = 10 where x = 4/5
  • 13x – 10 = 150 where x = 160/13

The standard formula for linear inequalities in one variable is ax + b = 0 where a and b are integers, x is the variable, and the value of a and b can never be equal to 0.

Solving inequalities in one variable – basic rules for solving 

The application of formulae and basic rules for solving inequalities help in solving mathematical expressions quickly during entrance exams. Here are some rules to consider while solving the linear inequalities class 11 problems.

  • Adding or subtracting both sides of inequality with an equal number does not change or alter the sign
  • On dividing or multiplying both sides of the linear inequality expression by a positive number without any impact on the sign
  • The division or multiplication of both sides of the linear inequality expression by a negative number leads to the reversal of the linear inequality sign.

Linear inequalities in one variable – formulas for solving the equations 

The concept of linear equations and inequalities in one variable is first introduced in 6th Grade Mathematics, and gradually, there is a progression through the successive grades from 6th to 12th grade. However, the formulae remain the same, but the complexity of the problem gradually increases. In the CBSE Class 11, the topic consists of the integration of algebra concepts, algebra derivatives, algebra of real numbers, etc., as higher modules of a linear inequality. Here are some formulae of linear inequality algebraic function important .

 Property

Formulas

Distributive Property

a (b +c) = a x b + a x c

Commutative Property of Addition

a + b = b + a

Commutative Property of Subtraction

a x b = b x a

Associative Property of Addition

a + (b + c) = (a + b) + c

Associative Property of Multiplication

a x (b x c) = (a x b) x c

Additive Identity Property

a + 0 + a

Multiplicative Identity Property

a x 1 = a

Additive Inverse Property

a + ( – a) = 0

Multiplicative Inverse Property

a (1/a) = 1

Zero Property of Multiplication

a x (0) = 0 <

Linear inequalities in one variable – some examples for better understanding

  1. Solve the following equation: 6x + 9= 18x – 12

Solution: In this, first shift the values of the variables on one side. Shift -12 on the left side expression, and as it is shifting sides, there will be a sign value change from – to + and shift 6x to right-hand side with a sign change. The equation after shift will be:

Step 1

6x + 9= 18x – 12

9 = 18x – 6x – 12

9 = 12x – 12

21 = 12x

Step 2: Divide the equation by 12 on both sides to verify the equation.

21/12 = x

Step 3: Substitute the value of x in the equation.

6x + 9= 18x – 12

6 x 21/12 + 9 = 18 x 21/12 – 12

39/2 = 39/2 (Hence proved)

  1. Solve 5x – 3 =7, when
  2. i) x is an integer
  3. ii) x is a real number

Solution:

The given inequality equation is 5x – 3 = 7

Now, add 3 to both sides of the equation

= 5x – 3 + 3 = 7 +3

= 5x =10 (Now divide both sides by 5)

= 5x/5 = 10/5

= x = 2

Hence it is proved that

  1. i) The integer is less than 2 where x is an integer and (ii) x is a real number and as x = 2, it can be all the real numbers which are less than 2.

Conclusion

The first foundation for understanding the concepts of linear equations and inequalities in one variable is from 6th grade onwards. The linear inequalities in one variable are an important concept for solving different types of mathematical questions and for preparing for various entrance exams. It can be used for calculating the value of an integer when two expressions are given. Knowing the formulae and using the expression of equation form helps find the value of an integer. The use of linear equations and inequality helps in reducing problems related to integers, algebra, and some arithmetic mathematical problems.