Chapter Notes: Linear Inequations

Table of Contents
1. Introduction
2. Linear Inequalities in One Variable
3. Solving a Linear Inequation Algebraically
4. Replacement Set and Solution Set
5. Representation of the Solution on the Number Line
6. Combining Inequations
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Introduction

Imagine you're planning a budget for a school trip, and you need to ensure your expenses stay within a certain limit. Or perhaps you're trying to figure out how many hours you can study to balance fun and work without exceeding your available time. This is where linear inequations come into play! In mathematics, linear inequations help us describe relationships where quantities aren't exactly equal but are greater than, less than, or equal to each other. This chapter, designed for Class 10 ICSE Mathematics, dives into the world of linear inequations in one variable, teaching us how to solve them, understand their solutions, and represent them visually on a number line. It's like solving a puzzle that helps us make sense of real-world constraints in a clear and logical way!

• An inequation is a statement that compares two quantities using inequality signs: > (greater than), ≥ (greater than or equal to), < (less than), or ≤ (less than or equal to).
• Unlike equations, inequations show a range of possible values rather than a single solution.
• Inequations are used to represent conditions where one quantity is not exactly equal to another.

Example: If x and y are two quantities, they can satisfy conditions like x > y, x ≥ y, x < y, or x ≤ y. For instance, x < 8 is an inequality stating x is less than 8.

Linear Inequalities in One Variable

• A linear inequation in one variable involves a single variable (like x) with a linear expression (no exponents higher than 1).
• It compares the linear expression to a constant using inequality signs.
• Forms of linear inequalities:
  • ax + b > c: ax + b is greater than c.
  • ax + b < c: ax + b is less than c.
  • ax + b ≥ c: ax + b is greater than or equal to c.
  • ax + b ≤ c: ax + b is less than or equal to c.
• The signs >, <, ≥, and ≤ are called signs of inequality.
• Example: 3x + 5 > 8 is a linear inequation where we need to find values of x that make the expression 3x + 5 greater than 8.

Solving a Linear Inequation Algebraically

• Solving a linear inequation means finding all possible values of the variable that satisfy the inequation.
• The process involves isolating the variable using algebraic operations while maintaining the inequality sign.

Working Rules for Solving Linear Inequations:

Rule 1: When transferring a positive term to the other side, its sign becomes negative.

Rule 2: When transferring a negative term to the other side, its sign becomes positive.

Rule 3: Multiplying or dividing both sides by the same positive number does not change the inequality sign.

Rule 4: Multiplying or dividing both sides by the same negative number reverses the inequality sign.

Rule 5: Changing the sign of every term on both sides reverses the inequality sign.

Rule 6: If both sides of an inequation are positive or both negative, taking their reciprocals reverses the inequality sign.

Replacement Set and Solution Set

• Replacement Set: The set from which values of the variable are chosen.
• Solution Set: The subset of the replacement set whose elements satisfy the inequation.
• The solution set depends on the type of replacement set (natural numbers, whole numbers, integers, or real numbers).

Example 1: For x < 3:

• If replacement set is natural numbers (N): Solution set = {1, 2}
• If replacement set is whole numbers (W): Solution set = {0, 1, 2}
• If replacement set is integers (Z): Solution set = {..., -2, -1, 0, 1, 2}
• If replacement set is real numbers (R): Solution set = {x : x < 3, x ∈ R}

Example 2: Solve 3x + 4 < 16, where the replacement set is natural numbers (N).

• Step 1: 3x + 4 < 16
• Step 2: Subtract 4: 3x < 16 - 4 → 3x < 12
• Step 3: Divide by 3: x < 12/3 → x < 4
• Since x ∈ N, solution set = {1, 2, 3}

Alternative Method Example:
Solve 8 - x ≤ 4x - 2, where the replacement set is the natural numbers (N).

Method 1:

• Step 1: 8 - x ≤ 4x - 2
• Step 2: Subtract 4x: 8 - x - 4x ≤ -2 → 8 - 5x ≤ -2
• Step 3: Subtract 8: -5x ≤ -2 - 8 → -5x ≤ -10
• Step 4: Divide by -5 (reverse inequality): x ≥ 2
• Solution set = {2, 3, 4, 5, ...}

Alternative Method:

• Step 1: Rewrite as 4x - 2 ≥ 8 - x
• Step 2: Add x: 4x + x - 2 ≥ 8
• Step 3: Add 2: 5x ≥ 10
• Step 4: Divide by 5: x ≥ 2
• Solution set = {2, 3, 4, 5, ...}

Representation of the Solution on the Number Line

Solutions of inequations can be represented on a real number line.

Convention:

• Use a hollow circle (◦) for strict inequalities (< or >) to show the endpoint is not included.
• Use a filled circle (•) for inclusive inequalities (≤ or ≥) to show the endpoint is included.

Example 1:
For x < 2, x ∈ R:

• Draw a number line with a hollow circle at 2 and an arrow pointing left to show all numbers less than 2.
• Graph: -5 -4 -3 -2 -1 0 1 2◦ 3 4 5 (arrow left from 2)

For x ≤ 2, x ∈ R:

• Draw a number line with a filled circle at 2 and an arrow pointing left.
• Graph: -5 -4 -3 -2 -1 0 1 2• 3 4 5 (arrow left from 2)

Example 2:
Solve and graph -2 < x ≤  4, y ∈ Z.

• Step 1: The inequality means x > -2 and x ≤  4.
• Step 2: Since x is an integer, solution set = { -1, 0, 1, 2, 3, 4}.
• Step 3: On the number line, place a filled circle at -1 and a filled circle at 4.

Combining Inequations

• Combining inequalities involves finding the solution set that satisfies multiple inequations, either their intersection (AND) or union (OR).
• Intersection (AND): Common elements that satisfy both inequations (A ∩ B).
• Union (OR): Elements that satisfy at least one of the inequations (A ∪ B).

Example 1: Solve and graph 3x + 6 ≥ 9 and -5x > -15, x ∈ R.

For 3x + 6 ≥ 9:

• Step 1: Subtract 6: 3x ≥ 3
• Step 2: Divide by 3: x ≥ 1

For -5x > -15:

• Step 1: Divide by -5 (reverse inequality): x < 3

• Combined: 1 ≤ x < 3

Example 2: Solve and graph -2 < 2x - 6 or -2x + 5 ≥ 13, x ∈ R.

For -2 < 2x - 6:

• Step 1: Add 6: 4 < 2x
• Step 2: Divide by 2: x > 2

For -2x + 5 ≥ 13:

• Step 1: Subtract 5: -2x ≥ 8
• Step 2: Divide by -2 (reverse inequality): x ≤ -4

• Combined (union): x ≤ -4 or x > 2

Important

Product of Two Numbers:

• If the product of two numbers is negative (< 0), one number is positive, and the other is negative.
• If the product of two numbers is positive (> 0), both numbers are positive or both are negative.

Example for Negative Product:
Solve (x - 3)(x + 5) < 0, x ∈ R.

• Step 1: The product is negative, so one factor is positive, and the other is negative.
• Case 1: x - 3 > 0 and x + 5 < 0
• x > 3 and x < -5 (not possible, as x cannot be both > 3 and < -5).
• Case 2: x - 3 < 0 and x + 5 > 0
• x < 3 and x > -5 → -5 < x < 3
• Solution set: {x : -5 < x < 3, x ∈ R}

Example for Positive Product:
Solve (x - 3)(x + 5) > 0, x ∈ R.

• Step 1: The product is positive, so both factors are positive or both are negative.
• Case 1: x - 3 < 0 and x + 5 < 0
• x < 3 and x < -5 → x < -5
• Case 2: x - 3 > 0 and x + 5 > 0
• x > 3 and x > -5 → x > 3
• Combined (union): x < -5 or x > 3