Login / Register
Follow Us
0
0
Register Free!
Register Free!
Online Mock Tests & Exam Preparation | Target Exams
Select Medium / माध्यम चुनें:
Eng (English) Hindi (हिन्दी)
ICSE • Class 7 • Mathematics • Ch 19
Estimated Time: 45 Mins
Study Progress: In Progress

Linear Inequations

In ICSE Class 7 Mathematics, &quot;Linear Inequations&quot; provides an authoritative, algebraic master study guide analyzing the solution and graphical representation of mathematical inequality relations. This comprehensive chapter explores Concept of an Inequation (Statements involving inequality symbols: $, \ge$; Linear inequation in one variable of the form $ax + b < c$), The Replacement Set / Domain ($\mathbb{N}$ [Natural numbers], $\mathbb{W}$ [Whole numbers], $\mathbb{Z}$ [Integers], $\mathbb{R}$ [Real numbers]), The Solution Set (The subset of the replacement set whose elements satisfy the inequation), Fundamental Rules of Inequality (Rule 1: Adding or subtracting the same number on both sides preserves the inequality sign; Rule 2: Multiplying or dividing by a positive number preserves the inequality sign; CRITICAL Rule 3: Multiplying or dividing both sides by a NEGATIVE number REVERSES the inequality sign: $a < b \implies -a > -b$), Solving Inequations by Transposition, and Graphing Solution Sets on a Real Number Line (Representing discrete elements using bold solid dots $\bullet$ for $\mathbb{N}, \mathbb{W}, \mathbb{Z}$; Open circle $\circ$ for strict inequalities on $\mathbb{R}$, closed circle $\bullet$ for inclusive inequalities, and shaded bold rays) aligned with the 2026–27 CISCE curriculum.

Take Chapter CBT Test View All Chapters
Overview Core Concepts Formulas & Identities Summary Self Check Take MCQ Test

Why Does Multiplying Both Sides of an Inequality by Minus One Completely Flip the Direction of the World Like a Magic Mirror?

Consider two simple numbers: $3$ and $5$. Everyone knows that $3 < 5$. But now, multiply both sides by $-1$: does $-3$ remain less than $-5$? Look at the integer number line: $-3$ lies to the RIGHT of $-5$, which means $-3 > -5$! The inequality sign completely flipped around! This is the most dangerous, fatal trap in all of algebra: whenever you multiply or divide an inequation by a negative number, the inequality sign MUST REVERSE! While a linear equation ($2x = 6$) has only one single unique root ($x = 3$), a linear inequation ($2x < 6$) opens up a boundless universe of solutions! But your final answer depends entirely on your "Replacement Set": if you are hunting in the realm of Natural Numbers ($\mathbb{N}$), the solutions are $\{1, 2\}$; if you are in Integers ($\mathbb{Z}$), they are $\{\dots, -2, -1, 0, 1, 2\}$; and if you are in Real Numbers ($\mathbb{R}$), there are infinite numbers plotted with a shaded arrow on a number line! What is the difference between an open circle ($\circ$) and a solid dot ($\bullet$) on a number line? Let's master linear inequations.

Why This Chapter Matters

Inequalities are essential in computer programming (conditional IF statements, loop bounds), business operations research (maximizing profit under budget constraints), speed limits, credit limits, and engineering safety margins. Mastering inequation sign reversals and number-line plotting is a signature ICSE mathematics requirement.

Before You Begin (Prerequisites)

  • Operations on integers and sign rules from Chapter 1.
  • Solving linear equations by transposition from Chapter 10.
  • Plotting numbers on the number line.

What You Will Learn (Core Objectives)

  • Define an inequation and identify inequality symbols ($<, \le, >, \ge$).
  • Differentiate between the Replacement Set (Domain) and the Solution Set.
  • Apply the golden rule of inequality reversal when multiplying or dividing by negative numbers.
  • Solve linear inequations in one variable using systematic transposition.
  • Graph solution sets on the number line for $\mathbb{N}, \mathbb{W}, \mathbb{Z}$, and $\mathbb{R}$ using open/solid dots.
  • Solve multi-step fractional linear inequations with double inequalities ($a < x \le b$).

Chapter Roadmap & Progression

1 1. Concept of an Inequation & The F...
2 2. Fundamental Axioms of Inequality...
3 3. Solving Inequations by Transposi...
4 4. Graphing Solution Sets on the Nu...

Complete Concept Guide (100% Curriculum Coverage)

1. Concept of an Inequation & The Four Inequality Symbols

Understand
A. What is an Inequation?

An Inequation (or Inequality) is an algebraic statement containing one of the four inequality relation symbols:

  • $<$ (Strictly Less Than): $x < 5$ (5 is not included).
  • $\le$ (Less Than or Equal To): $x \le 5$ (5 is included).
  • $>$ (Strictly Greater Than): $x > -2$ (-2 is not included).
  • $\ge$ (Greater Than or Equal To): $x \ge -2$ (-2 is included).
B. Replacement Set vs Solution Set:
  • Replacement Set (Domain): The overarching set of numbers from which values of the variable $x$ may be chosen (e.g., $\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{R}$).
  • Solution Set: The set of all values chosen from the replacement set that make the inequation a true mathematical statement.

2. Fundamental Axioms of Inequality (The Reversal Rule)

Inequality Rules

Let $a, b, c \in \mathbb{R}$:

  1. Addition / Subtraction Rule: Adding or subtracting the same number on both sides leaves the inequality sign UNCHANGED: $$a < b \implies a + c < b + c \quad \text{and} \quad a - c < b - c$$
  2. Positive Multiplication / Division Rule: Multiplying or dividing both sides by a positive number ($c > 0$) leaves the sign UNCHANGED: $$a < b \implies ac < bc \quad \text{and} \quad \frac{a}{c} < \frac{b}{c}$$
  3. THE GOLDEN RULE OF INEQUATIONS (Negative Reversal):

    Multiplying or dividing both sides of an inequation by a NEGATIVE NUMBER ($c < 0$) REVERSES the inequality sign:

    $$\mathbf{a < b \implies ac > bc} \quad \text{and} \quad \mathbf{\frac{a}{c} > \frac{b}{c}}$$

    Numerical Proof: We know $2 < 5$. Multiply by $-1$: $-2 > -5$ (since $-2$ lies to the right of $-5$). The sign flipped!

3. Solving Inequations by Transposition

Transposition

To solve a linear inequation:

  1. Clear fractions by multiplying by the positive LCM of denominators.
  2. Collect variable terms on one side (usually LHS) and constant terms on the other side (RHS) using transposition.
  3. Divide by the coefficient of $x$. If the coefficient is negative, flip the inequality sign immediately!
  4. Write the final Solution Set filtered through the given Replacement Set.

4. Graphing Solution Sets on the Number Line

Number Line Graphing
A. For Discrete Sets ($\mathbb{N}, \mathbb{W}, \mathbb{Z}$):

Plot a bold solid dot ($\\bullet$) exclusively over the specific discrete integers that belong to the solution set. Do not draw continuous lines between dots!

B. For Continuous Real Numbers ($\mathbb{R}$):
  • Open Circle ($\\circ$): Drawn at an endpoint if that number is excluded ($<$ or $>$).
  • Closed Solid Circle ($\\bullet$): Drawn at an endpoint if that number is included ($\le$ or $\ge$).
  • A thick shaded line or arrow is drawn over the continuous line segment representing all real numbers.

Key Formulas, Identities & Theorems

Golden Inequation Reversal Law
$$a < b \iff -a > -b \quad (\text{Multiplying or dividing by } -1)$$
Sign of inequality reverses on negative multiplication/division.
Replacement-Solution Set Inclusion
$$\text{Solution Set} \subseteq \text{Replacement Set}$$
Solutions must be elements of the specified domain.

Linear Inequations: Reversal Rule & Number Line Graphs

Linear Inequations: The Reversal Law & Number Line Graphing THE GOLDEN REVERSAL LAW • Multiplying/Dividing by a NEGATIVE: a < b ⇔ -a > -b Example: 3 < 5 ⇒ -3 > -5 (Sign Flips!) • Adding or Subtracting: Sign UNCHANGED • Multiplying by POSITIVE: Sign UNCHANGED • Replacement Sets: ℕ={1,2,...} • 𝕎={0,1,...} • ℤ={...,-1,0,1,...} • ℝ • Solution Set ⊆ Replacement Set NUMBER LINE GRAPHING 1. Discrete Sets (ℤ, 𝕎, ℕ): Plot individual BOLD DOTS (•) only! 1 2 3 2. Real Numbers (ℝ): Shaded continuous line/arrow 1 (Solid • = ≥) → ∞ • Open circle (ο) for < or > (excluded) CRITICAL LAW: MULTIPLYING OR DIVIDING BY A NEGATIVE REVERSES THE INEQUALITY SIGN!

Chapter Summary & 10 Key Takeaways

Takeaway 1
An inequation contains inequality relation symbols: $<, \le, >, \ge$.
Takeaway 2
Replacement Set (Domain) is the universe from which the variable may be chosen.
Takeaway 3
Solution Set is the collection of elements from the replacement set that satisfy the inequation.
Takeaway 4
Adding or subtracting the same number on both sides preserves the inequality sign.
Takeaway 5
Multiplying or dividing by a positive number preserves the inequality sign.
Takeaway 6
CRITICAL: Multiplying or dividing by a NEGATIVE number REVERSES the inequality sign ($a < b \implies -a > -b$).
Takeaway 7
For discrete domains ($\mathbb{N}, \mathbb{W}, \mathbb{Z}$), plot isolated solid dots $\bullet$ on the number line.
Takeaway 8
For continuous domains ($\mathbb{R}$), use an open circle $\circ$ for strict inequality ($<, >$) and closed dot $\bullet$ for inclusive ($\le, \ge$).
Takeaway 9
A double inequation $a \le x < b$ denotes elements bounded simultaneously between $a$ and $b$.
Takeaway 10
Always filter the final algebraic inequality through the specific Replacement Set given in the problem.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
Solve the inequation: $3 - 2x \ge x - 12$, where the replacement set is $x \in \mathbb{W}$ (Whole numbers). Write the solution set.
Reveal Answer & Explanation
Answer:

Step 1: Transpose $x$ to LHS and $3$ to RHS:

$$3 - 2x \ge x - 12$$


$$-2x - x \ge -12 - 3$$


$$-3x \ge -15$$


Step 2: Divide both sides by $-3$. REVERSE the inequality sign from $\ge$ to $\le$:

$$x \le \frac{-15}{-3} \implies \mathbf{x \le 5}$$


Step 3: Filter by Replacement Set $\mathbb{W} = \{0, 1, 2, 3, \dots\}$:

$$\text{Solution Set} = \mathbf{\{0, 1, 2, 3, 4, 5\}}$$

.


Transpose to get $-3x \ge -15$. Dividing by $-3$ reverses the sign to $x \le 5$. Whole numbers are $\{0, 1, 2, 3, 4, 5\}$.
2
Solve: $\frac{x}{2} - 5 < \frac{x}{3} - 4$, given that $x \in \mathbb{Z}$ (Integers). What is the greatest integer value of $x$?
Reveal Answer & Explanation
Answer:

Step 1: Multiply all terms by the LCM of denominators ($2, 3$), which is $6$:

$$6 \left( \frac{x}{2} \right) - 6(5) < 6 \left( \frac{x}{3} \right) - 6(4)$$


$$3x - 30 < 2x - 24$$


Step 2: Transpose $2x$ to LHS and $-30$ to RHS:

$$3x - 2x < -24 + 30$$


$$\mathbf{x < 6}$$


Step 3: Since $x \in \mathbb{Z}$, the solution set is $\{\dots, 2, 3, 4, 5\}$.
• The greatest integer value of $x$ is $5$.


Multiply by 6 to get $3x - 30 < 2x - 24 \implies x < 6$. The greatest integer less than 6 is 5.
3
Solve the double inequation: $-3 \le 2x - 1 < 5$, given $x \in \mathbb{R}$ (Real numbers). Represent the solution set on a number line.
Reveal Answer & Explanation
Answer:

Step 1: Add $1$ to all three parts of the inequality:

$$-3 + 1 \le 2x - 1 + 1 < 5 + 1$$


$$-2 \le 2x < 6$$


Step 2: Divide all parts by $2$ (positive, sign unchanged):

$$\frac{-2}{2} \le \frac{2x}{2} < \frac{6}{2}$$


$$\mathbf{-1 \le x < 3}$$


Step 3: Number Line Representation:
• Draw a solid closed circle ($\\bullet$) at $-1$ (included).
• Draw an open circle ($\\circ$) at $3$ (excluded).
• Draw a thick, bold shaded line segment connecting $-1$ to $3$.


Add 1 to all sides ($-2 \le 2x < 6$), divide by 2 ($-1 \le x < 3$). Plot solid dot at -1, open circle at 3.
4
If the replacement set is $\mathbb{N} = \{1, 2, 3, 4, \dots\}$, find the solution set of: $4(x - 1) \le 3(x + 1) + 2$.
Reveal Answer & Explanation
Answer: Step 1: Expand brackets using the Distributive Law:
$$4x - 4 \le 3x + 3 + 2$$
$$4x - 4 \le 3x + 5$$
Step 2: Transpose $3x$ to LHS and $-4$ to RHS:
$$4x - 3x \le 5 + 4$$
$$\mathbf{x \le 9}$$
Step 3: Filter by Replacement Set $\mathbb{N}$ (Natural numbers start from $1$):
$$\text{Solution Set} = \mathbf{\{1, 2, 3, 4, 5, 6, 7, 8, 9\}}$$.
Expand to $4x - 4 \le 3x + 5 \implies x \le 9$. Natural numbers start at 1, so the solution is $\{1, 2, \dots, 9\}$.
5
Explain with a mathematical proof why multiplying an inequation by a negative number reverses the inequality symbol.
Reveal Answer & Explanation
Answer:

• Let $a$ and $b$ be real numbers such that $a < b$.
• Subtract $b$ from both sides:

$$a - b < 0$$


• Subtract $a$ from both sides:

$$-b < -a \implies \mathbf{-a > -b}$$


Notice that starting from $a < b$, we arrived at $-a > -b$.
Therefore, multiplying both sides by $-1$ reverses the inequality symbol from $<$ to $>$.


Subtract $(a+b)$ from both sides: $a - (a+b) < b - (a+b) \implies -b < -a \implies -a > -b$.
6
Solve: $7 - 3x > -2$, given that $x \in \mathbb{Z}^+$ (Positive integers).
Reveal Answer & Explanation
Answer:

Step 1: Transpose $7$ to RHS:

$$-3x > -2 - 7$$


$$-3x > -9$$


Step 2: Divide by $-3$ and reverse the inequality sign:

$$x < \frac{-9}{-3} \implies \mathbf{x < 3}$$


Step 3: Filter by Replacement Set $\mathbb{Z}^+ = \{1, 2, 3, \dots\}$:

$$\text{Solution Set} = \mathbf{\{1, 2\}}$$

.


$-3x > -9 \implies x < 3$. Positive integers strictly less than 3 are $\{1, 2\}$.
7
On a number line, what is the graphical difference between representing $x > 4$ and $x \ge 4$ on the set of real numbers $\mathbb{R}$?
Reveal Answer & Explanation
Answer:

• For $x > 4$ (Strict Inequality): An open circle ($\\circ$) is drawn at $4$ to indicate that the number $4$ itself is excluded from the solution set, with a bold shaded arrow extending infinitely to the right.
• For $x \ge 4$ (Inclusive Inequality): A solid, filled-in dot ($\\bullet$) is drawn at $4$ to signify that $4$ is included in the solution set, with a bold shaded arrow extending infinitely to the right.


Open circle ($\circ$) excludes the endpoint ($>$); solid dot ($ullet$) includes the endpoint ($\ge$).
8
Find the smallest integer $x$ that satisfies the inequation: $5x - 4 > 2(x + 1)$.
Reveal Answer & Explanation
Answer:

Step 1: Expand brackets:

$$5x - 4 > 2x + 2$$


Step 2: Transpose $2x$ to LHS and $-4$ to RHS:

$$5x - 2x > 2 + 4$$


$$3x > 6$$


$$x > \frac{6}{3} \implies \mathbf{x > 2}$$


Since $x \in \mathbb{Z}$ and $x$ must be strictly greater than $2$, the smallest integer is $3$.


$3x > 6 \implies x > 2$. The smallest integer strictly greater than 2 is 3.
Finished Studying This Chapter?
READY TO PRACTICE?

Timed CBT Practice Tests (Exam Simulator)

Put your concepts to the test with official curriculum-aligned Foundation and Advanced practice tests. Get instant accuracy scores, time metrics, and step-by-step verified explanations.

Foundation Test
15 MCQs • Core Concepts
Advanced Test
15 MCQs • HOTS & Traps

All Class 7 Mathematics Chapters

AI Study Friend

Instant Concept Tutor

Have a doubt in Linear Inequations? Ask our AI study tutor for rapid explanations or customized quizzes.