Linear Inequations Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

A linear inequality is a mathematical expression involving a linear function and an inequality symbol such as <, >, ≤, or ≥. Unlike equations, which show equality, inequalities represent a range of values that satisfy the given condition. These are commonly used to describe constraints or conditions in various mathematical and real-world problems.
In this chapter, we focus on solving linear inequalities and representing their solutions, often using number lines, with specific replacement sets such as natural numbers (N), whole numbers (W), integers (Z), or real numbers (R).

Key Concepts

1. Inequality Symbols

Linear inequalities use the following symbols:

• <: Less than
• >: Greater than
• ≤: Less than or equal to
• ≥: Greater than or equal to

Example 1

Q: Identify the inequality symbol in 2x + 3 ≥ 7.
Sol: The inequality symbol is ≥ (greater than or equal to).

2. Replacement Set and Solution Set

Replacement Set: The set of all possible values that a variable can take. For example, natural numbers (N = {1, 2, 3, ...}), whole numbers (W = {0, 1, 2, ...}), integers (Z = {..., -2, -1, 0, 1, 2, ...}), or a specific finite set like {-5, -2, 0, 2, 5}.
Solution Set: The subset of the replacement set that satisfies the inequality.

Example 1

Q: Solve x + 2 < 5, x ∈ N.
Sol: x + 2 - 2 < 5 - 2 (Subtract 2)
x < 3
Since x ∈ N, solution set = {1, 2}.

Example 2

Q: Solve 2x > 4, x ∈ {-5, -2, 0, 2, 5}.
Sol: 2x > 4
x > 2 (Divide by 2)
From the replacement set {-5, -2, 0, 2, 5}, solution set = {5}.

3. Solving Linear Inequalities

Solving a linear inequality involves finding all values of the variable that make the inequality true. The process is similar to solving linear equations, with some additional rules:

• Addition/Subtraction: Adding or subtracting the same number from both sides does not affect the inequality direction.
• Multiplication/Division by a Positive Number: Multiplying or dividing both sides by a positive number does not change the inequality direction.
• Multiplication/Division by a Negative Number: Multiplying or dividing both sides by a negative number reverses the inequality direction.

Note: When dividing or multiplying by a negative number, reverse the inequality sign. For example, if -2x > 8, dividing by -2 gives x < -4.

Example 1

Q: Solve 4x - 3 > 9, x ∈ W.
Sol: 4x - 3 + 3 > 9 + 3 (Add 3)
4x > 12
x > 3 (Divide by 4)
Since x ∈ W, solution set = {3, 4, 5, ...}.

Example 2

Q: Solve -2x ≤ 6, x ∈ Z.
Sol: -2x ≤ 6
x ≥ -3 (Divide by -2, reverse inequality)
Since x ∈ Z, solution set = {-3, -2, -1, 0, 1, 2, ...}.

4. Representing Solutions on a Number Line

Solutions to inequalities can be represented on a number line to visualize the range of values:

• Use an open circle (○) for < or > to indicate the endpoint is not included.
• Use a closed circle (●) for ≤ or ≥ to indicate the endpoint is included.
• Draw an arrow to show the direction of the solution set.

Example 1

Q: Graph x > 2, x ∈ Z.
Sol: On a number line, place an open circle at 2 and draw an arrow to the right.

... 0 1 ○ 2 3 4 ...
             →

Example 2

Q: Graph x ≤ 4, x ∈ W.
Sol: Place a closed circle at 4 and draw an arrow to the left

... 0 1 2 3 ● 4 ...
←

Solving Linear Inequalities: Step-by-Step

Below are the general steps to solve linear inequalities:

1. Simplify the Inequality: Expand any brackets and combine like terms.
2. Isolate the Variable: Use addition, subtraction, multiplication, or division to get the variable on one side.
3. Consider the Replacement Set: List only the values from the replacement set that satisfy the inequality.
4. Graph the Solution (if required): Represent the solution set on a number line.

Example 1

Q: Solve 3(x -- 1) > 6, x ∈ N.
Sol: 3x - 3 > 6
3x > 9 (Add 3)
x > 3 (Divide by 3)
Since x ∈ N, solution set = {4, 5, 6, ...}.

Example 2

Q: Solve (x/2) + 1 ≤ 3, x ∈ W.
Sol: (x/2) ≤ 2 (Subtract 1)
x ≤ 4 (Multiply by 2)
Since x ∈ W, solution set = {0, 1, 2, 3, 4}.