Linear Inequation in One Variable states that one quantity is not equal to another quantity which having one variable. For example, if ax + b = 0 where x is the variable and a, b are the fixed numbers or integers. We do the comparison for the expression to find them as inequations. We use inequality symbols like <, >, ≤ or ≥ to compare the expressions or values. One Variable in Linear Inequations especially explains about the single variable only.

## Definition of Linear Inequation in One Variable

Linear Inequalities are the expressions that are not equal to each other when compared between two values. Linear Inequation in One Variable represents the comparison between expressions those having a single variable are not equal when compared between two values. The symbols that represent inequalities are <, >, ≥, ≤.

Let us consider two quantities p and q which are not equal to each other p ≠ q. The linear inequalities will satisfy the following conditions. They are

(i) p > q

(ii) p ≥ q

(iii) p < q

Or, p ≤ q

All of the above conditions belong to inequation.

Also, Check:

- Properties of Inequation or Inequalities
- Linear Inequality and Linear Inequations
- Representation of the Solution Set of a Linear Inequation

### Statements that obey One Variable Linear Inequation

Let us take a variable m. m is a variable that when added to 3 gives a sum less than 7.

The above sentence can show as m + 3 < 7, where < is the less than symbol and 3 and 7 are integers. m is the one variable in the linear inequation m + 3 < 7.

m is less than 3. Because adding numbers less than 3 to m gives you the number that is less than 7. We can also write the expression as m < 4.

### Linear Inequation Form

The structure of a linear inequation in one variable is ax + b < c where x is the variable and a, b, and c are fixed numbers belonging to the set R. If you consider a, b and c as real numbers, then all of the below are treats as linear inequation in one variable.

- ax + b ≥ c (‘≥’ stands for “is greater than or equal to”)
- ax + b > c (‘>’ stands for “is greater than”)
- ax + b ≤ c (‘≤’ stands for “is less than or equal to”) are linear inequation in one variable where the signs in inequations, ‘>’, ‘<’, ‘≥’ and ‘≤’ are known as the signs of inequality.

### Important Notes on Linear Inequation in One Variable

Let us take two real numbers a and b, then

- a is less than b, written as a < b, if and only if b – a is positive. For example,

(i) 4 < 6, where 6 – 4 = 2 which is positive in number.

(ii) -7 < -5, where -5 – (-7) = -5 + 7 = 2 which is positive in number.

(iii) \(\frac { 4 }{ 6 } \) < \(\frac { 8 }{ 10 } \), where \(\frac { 8 }{ 10 } \) – \(\frac { 4 }{ 6 } \) = \(\frac { 2 }{ 15 } \) which is a positive number. - a is less than or equal to b, then we can write it as a ≤ b, if and only if a – b is either zero or positive. For example,

(i) -8 ≤ 14, since 14 – (-8) = 14 + 8 = 22 which is positive number.

(ii) \(\frac { 2 }{ 3 } \) ≤ \(\frac { 2 }{ 3 } \), where \(\frac { 2 }{ 3 } \) – \(\frac { 2 }{ 3 } \) = 0 - a is greater than or equal to b, then we can write it as a ≥ b, if and only if a – b is either zero or positive. For example,

(i) 8 ≥ -12, since 8 – (-12) = 8 + 12 = 20 which is positive number.

(ii) \(\frac { 10 }{ 16 } \) ≥ \(\frac { 10 }{ 16 } \), where \(\frac { 10 }{ 16 } \) – \(\frac { 10 }{ 16 } \) = 0. - a is greater than b, then we can write it as a > b, if and only if a – b is positive. For example,

(i) 6 > 4, where 6 – 4 = 2 which is a positive number.

(ii) -7 > -11, where -7 – (-11) = – 7 + 11 = 4 which is a positive number.

(iii) \(\frac { 8 }{ 10 } \) > \(\frac { 4 }{ 6 } \), where \(\frac { 8 }{ 10 } \) – \(\frac { 4 }{ 6 } \) = \(\frac { 2 }{ 15 } \) which is a positive number.

### Algebraic Solutions of Linear Inequalities in One Variable

Let us discuss how to solve the inequalities using the below example.

**Example:** 6x < 12, where x is the whole number.

LHS of the inequality is 6x and RHS of the inequality is 12.

When x = 0, LHS of the inequality is 6 × 0 = 0, RHS of the inequality = 12.

Since 0 is less than 12, x = 0 satisfies the inequality.

When x = 1, LHS of the inequality is 6 × 1 = 6, RHS of the inequality = 12.

Since 1 is less than 12, x = 1 satisfies the inequality.

When x = 2, LHS of the inequality is 6 × 2 = 12, RHS of the inequality = 12.

Since 12 is not less than 12, x = 2 will not satisfy the inequality.

Therefore, the x should be less than 2 to satisfy the inequality. The solution set of the above inequality is 0 and 1 and {0, 1}.

It is the trial and error method that will take much time to solve the problem. We can follow some rules those will help the solving the inequalities.

### Rules for Solving Linear Inequations in One Variable

Check out the below rules for Linear Inequation in One Variable and note down them to have the perfect solution for problems.

**Rule 1**

Equal numbers may add or subtract from both sides of an inequality which will not affect the sign of inequality.

**Rule 2**

An inequality can multiply or divide by the same positive number on both sides. But when both sides of an inequality are multiplied or divided by a negative number, then the sign of inequality is reversed. For example, 6 > 4 where -6 < -4. Also, -16 < -14, whereas (-16)(-4) > (-14)(-4) i.e. 64 > 56.

#### Solving Linear Inequalities in One Variable Examples

**Example 1.**

6 – 4x < 2, solve for x if x is an integer less than 6.

**Solution:**

Given that 6 – 4x < 2.

Subtract 6 from both the sides, 6 – 6 – 4x < 2 – 6

-4x < -4

Divide -4 on both sides of the inequality.

As -4 is the negative number it is necessary to maintain a sign of inequality.

4x/-4 > -4/-4

x > 1.

Since x is an integer less than 6, the solution set of x is {2, 3, 4, 5}

### Representation of Linear Inequalities On Number Line

Learn the Representation of Linear Inequalities On Number Line using the below example. The Graphical Representation of Linear Inequalities having one variable shows only on Number Line as it has only one variable.

**Example:** Solve 4x + 6 ≤ 18, represent ‘x’ on a number line if x is positive.

**Solution:** Given that 4x + 6 ≤ 18 where x is the variable and remaining are real numbers.

Subtract 6 on both sides of the above inequality.

4x + 6 – 6 ≤ 18 – 6

4x ≤ 12

Divide the above inequality with 4 into both sides.

4x/4 ≤ 12/4

x ≤ 3

From the above figure, the numbers left side to the 3 satisfy the Linear Inequality in One Variable.

### Linear Inequalities in One Variable Word Problems

We have given Linear Inequalities in One Variable Word Problems to improve your preparation.

**Example 1.**

The marks obtained by a student of Class 7th in the first and second terminal examination are 50 and 35, respectively. Find the minimum marks he should get in the annual examination to have an average of at least 32 marks.

**Solution:**

Given that the marks obtained by a student of Class 7th in the first and second terminal examination are 50 and 35, respectively.

Let the x be the marks obtained by the student in the annual examination.

Now, the inequality equation becomes

(50 + 35 + x)/3 ≥ 52

Multiply 3 on both sides of the above inequality.

75 + x ≥ 156

Subtract 75 on both sides of the above equation.

75 – 75 + x ≥ 156 – 75

x ≥ 81

Therefore, the student must get a minimum of 81 marks to get an average of at least 52 marks in the annual exam.

**Example 2:**

Find all pairs of consecutive odd natural numbers, both of which are larger than 20, such that their sum is less than 80.

**Solution:**

Given that all pairs of consecutive odd natural numbers, both of which are larger than 20, such that their sum is less than 80.

Let x be the smaller of the two consecutive odd natural numbers so that the other one is x +2. Then, from the given information,

x > 20 ………….(1) and x + ( x + 2) < 80 ………..(2)

By solving the above equations, we get;

2x + 2 < 80

Subtract 2 on both sides of the above equation.

2x + 2 – 2 < 80 – 2

2x < 78

Divide 2 on both sides of the above equation.

2x/2 < 78/2

x < 39 ………….(3)

From eq.1 and eq.3 we get:

20 < x < 39

As x is an odd number, x can take the values 21, 23, 25, 27, 29, 31, 33, 35, and 37.

So, the required possible pairs will be (21, 23,), (23, 25), (25, 27), (27, 29), (29, 31), (31, 33), (33, 35), (35, 37).

### Frequently Asked Question and Answers on Linear Inequation in One Variable

**1. What are linear inequalities in one variable?**

linear inequalities in one variable are explained as the mathematical expression that consists of a single variable are compared with the help of the inequality symbol. Either the expression can be a numerical expression or an algebraic expression or a combination of both.

**2. What are the 5 different inequality symbols used in Maths?**

The 5 inequality symbols used to represent them are less than symbol (<), greater than symbol (>), less than or equal to a symbol (≤), greater than or equal to a symbol (≥), and not equal to a symbol (≠).

**3. How to solve inequality with one variable on both sides?**

Firstly, move the variables on one side and do the remaining operations like addition, subtraction, etc. If you do the multiplication or division, then change the inequality sign if the multiplication or division is done with -1.

**4. What is the similarity between linear equations and linear inequalities?**

The similarity between linear equations and linear inequalities is that they both are mathematical statements that relate two expressions to each other.