Solving Linear Inequalities In One Variable

Linear inequalities are algebraic or numerical expressions where two values are compared using inequality symbols. The form of linear inequalities differs from the form of linear equations. Linear equations have the general form of ax + b = c, where a ≠ 0. But when it comes to linear inequations, it can have one of four forms. 

• ax + b < c
• ax + b > c
• ax + b ≤ c
• ax + b ≥ c, where,

a, b, and c are real numbers. 

Since the expression has only one variable, it is an expression of linear inequalities in one variable. 

When you evaluate the unequal relationship between two algebraic expressions, which consists of two separate variables, it is an expression of linear inequalities in two variables.

Some examples of linear inequalities in two variables include:

• 3x < 2y + 2
• 5x > 3y + 7
• 6x + y + 4 ≤ 8y + 3
• x + 2y ≥ -9

Understanding linear inequalities

Example 1

Imagine you are sent to the grocery shop to buy sugar. Your mother gives you Rs. 300 and asks you to buy the maximum quantity possible. At the shop, the shopkeeper tells you that sugar is available at Rs. 70 per kg, and there are packets only of 1 kg each. 

Suppose the number of packets you purchase is x. The total amount you spend is 70x since the cost of each packet is Rs. 70. Now, you need to buy the maximum quantity possible with Rs. 300. You cannot spend the entire amount since 300 is not divisible by 70. Thus, you can express the statement as the following expression:

70x < 300 …. (1)

There is no equal sign in this expression. Hence, it is not an equation but an inequality. 

Example 2

You are buying cookies and juice at a shop. You have a budget of up to Rs. 150. The cost of one packet of cookies is Rs. 30, and a juice pack costs Rs. 20. 

To write this statement as an expression, take the number of cookies as x and the number of juice packs as y. The total amount you will spend is 30x + 20y. While 150 is divisible by 50 (30+20), you want to spend up to Rs. 150. Hence, 

30x + 20y ≤ 150 …. (2)

Notice this expression has two parts,

• 30x + 20y < 150 …. (3)
• 30x + 20y = 150 …. (4), where,

(3) is inequality, and (4) is an equation. 

Furthermore, since this expression involves two variables, x and y, you will need to solve the linear inequalities by applying the rules for two variables.

Solving linear inequalities in one variable – algebraic solutions

Continuing with the examples shared above, let us try to solve expression (1),

70x < 300

Here, x represents the number of packets purchased. So, you cannot take it as a fraction or a negative integer. The LHS = 70x and the RHS = 300. Using different values for x, let us find the solution to the inequation. 

• 70 (0) = 0, where x = 0. The answer is < 300, so the inequality is true
• 70 (1) = 70, where x = 1. The answer is < 300, so the inequality is true
• 70 (2) = 140, where x = 2. The answer is < 300, so the inequality is true 
• 70 (3) = 210, where x = 3. The answer is < 300, so the inequality is true
• 70 (4) = 280, where x = 4. The answer is < 300, so the inequality is true
• 70 (5) = 350, where x = 5. The answer is NOT < 300, so the inequality is false

Since the inequality is true for x = 0, 1, 2, 3, and 4, these are the solutions to the inequality. The set {0, 1, 2, 3, 4} are known as the solution set. 

If a value of the variable makes the inequality a true statement, it is a solution to the inequality in one variable.

To solve linear inequalities in class 11, here are the rules you need to follow:

1. Add or subtract equal numbers to/from both sides of the inequality.
2. Multiply or divide both sides of the inequality by the same non-zero number. Reverse the inequality sign in the case of multiplication or division with a negative number. 
3. Use the distributive property to remove the brackets. 
4. If there are fractions, get a common denominator. 

Graphing linear inequalities in one variable

For representing a linear inequality in a plane graphically, you need to follow the steps mentioned below:

1. If you need to represent ≥ or ≤, draw a thick graph line. The thickness of the graph line indicates that the solution set includes the points on this line. 
2. If you need to represent > or <, draw a dotted graph line. The dotted line indicates that the solution set does not include the points on the line. 
3. You can graphically represent the solutions of linear inequalities in one variable on the number line and the plane. But you can represent the solutions of two variables in the plane only.

Summing up

In this lesson, you learned about linear inequalities for Class 11. These equations do not have an equal sign. Instead, they use >, <, ≥, and ≤ signs to establish a relationship between two mathematical expressions. 

You need the same rules for solving linear inequalities in one variable as you need for solving linear equations with only one exception. Thus, you must apply the same formula to both sides of the inequality sign. The exception is that if you divide or multiply the terms with a negative number, the sign of inequality reverses. 

You need to use an open circle for > and < to graph linear inequalities. For ≥ and ≤, you use a closed circle.