Linear Inequations

A. The Golden Laws of Inequalities:

• Law 1 (Addition/Subtraction): Adding or subtracting the same quantity from both sides preserves the inequality sign: $$\text{If } a > b, \text{ then } a + c > b + c \quad \text{and} \quad a - c > b - c$$
• Law 2 (Positive Multiplication/Division): Multiplying or dividing both sides by a positive number preserves the inequality: $$\text{If } a > b \text{ and } c > 0, \text{ then } ac > bc \quad \text{and} \quad \frac{a}{c} > \frac{b}{c}$$
• Law 3: THE NEGATIVE INVERSION LAW (CRITICAL!): Multiplying or dividing both sides by a negative number REVERSES the inequality sign:
  $$\text{If } a > b \text{ and } c < 0, \text{ then } ac < bc \quad \text{and} \quad \frac{a}{c} < \frac{b}{c}$$ Example: $$-3x \le 9 \implies x \ge \frac{9}{-3} \implies x \ge -3$$

A. The Four Standard ICSE Replacement Sets:

Example: Suppose solving an inequation yields $x < 3.5$:

• If $x \in N$, Solution Set = $\{1, 2, 3\}$.
• If $x \in W$, Solution Set = $\{0, 1, 2, 3\}$.
• If $x \in Z$, Solution Set = $\{\dots, -2, -1, 0, 1, 2, 3\}$.
• If $x \in R$, Solution Set = $\{x : x < 3.5, x \in R\}$ (Continuous shaded segment).

A. The Splitting Protocol:

A compound double-inequation of the form $P \le Q < R$ must be split into two separate simultaneous inequations:

Solve both independently, and find their common intersection.

B. Worked Board Problem:

Problem: Solve the following inequation and represent the solution set on the number line:
$$-\frac{x}{3} - 4 \le \frac{x}{2} - 1 < \frac{1}{6}, \quad x \in R$$

Step-by-Step Solution:

Part 1: $-\frac{x}{3} - 4 \le \frac{x}{2} - 1$
Multiply through by $6$ (LCM of 3 and 2):
$$-2x - 24 \le 3x - 6 \implies -2x - 3x \le -6 + 24 \implies -5x \le 18$$ Divide by $-5$ (REVERSE INEQUALITY!):
$$x \ge -\frac{18}{5} \implies x \ge -3.6$$

Part 2: $\frac{x}{2} - 1 < \frac{1}{6}$
Multiply through by $6$:
$$3x - 6 < 1 \implies 3x < 7 \implies x < \frac{7}{3} \implies x < 2.33$$

Combining Solutions:
$$-3.6 \le x < 2.33, \quad x \in R$$ $$\text{Solution Set} = \{x : -3.6 \le x < 2\frac{1}{3}, x \in R\}$$

Problem:

Solve the following inequation and represent the solution set on the real number line:
$$-2 \le \frac{1}{2} - \frac{2x}{3} \le 1\frac{5}{6}, \quad x \in R$$

Step-by-Step Solution:

Express mixed fraction as improper fraction: $1\frac{5}{6} = \frac{11}{6}$.
$$-2 \le \frac{1}{2} - \frac{2x}{3} \le \frac{11}{6}$$ Multiply all three parts by $6$ (the LCM of 2, 3, 6):
$$6(-2) \le 6\left(\frac{1}{2} - \frac{2x}{3}\right) \le 6\left(\frac{11}{6}\right)$$ $$-12 \le 3 - 4x \le 11$$ Subtract $3$ from all three parts:
$$-12 - 3 \le -4x \le 11 - 3 \implies -15 \le -4x \le 8$$ Divide all parts by $-4$ (REVERSE ALL INEQUALITY SIGNS!):
$$\frac{-15}{-4} \ge \frac{-4x}{-4} \ge \frac{8}{-4}$$ $$3.75 \ge x \ge -2 \implies -2 \le x \le 3.75$$ Since $x \in R$, the Solution Set is:
$$\mathbf{\{x : -2 \le x \le 3.75, x \in R\}}$$ Number Line Graph: A thick continuous shaded bar starting at $-2$ with a dark solid circle and terminating at $3.75$ with a dark solid circle.

Rigorous Step-by-Step Solutions for Top-Band Scores:

Below is a curated compendium of standard ICSE board-level examination questions designed to test mathematical derivation, algebraic accuracy, unit precision, and rigorous geometric justifications.

How ICSE Examiners Award Marks in this Topic:

In the official CISCE evaluation rubrics, marks are systematically distributed across three distinct cognitive dimensions:

• Method Marks (M): Awarded for writing the correct formula, establishing the proper geometric theorem, setting up the correct equation, or constructing the proper table columns. Even if a careless calculation error occurs later, method marks are fully preserved!
• Accuracy Marks (A): Awarded for correct intermediate arithmetic steps, accurate factorization, algebraic simplification, and correct radical reduction.
• Final Statement & Unit Marks (B/A): Awarded for stating the final numerical answer with correct units, proper rounding (e.g. two decimal places), and answering all sub-parts explicitly.

Top 5 Practical Recommendations to Maximize Scores:

1. Never skip writing the standard formula: Always write out the formula in algebraic terms before substituting numerical values.
2. Keep rough calculations neatly organized: Draw a dedicated 2-inch rough margin on the right side of your answer sheet for long divisions and square root extractions.
3. Check boundary conditions: Ensure your final solutions belong strictly to the specified domain (e.g., natural numbers, positive lengths, valid quadrants).
4. State geometric reasons in parentheses: In geometry proofs, every equality must be accompanied by its supporting theorem name (e.g., '[angles in the same segment are equal]').
5. Proofread before moving to the next question: Spend 30 seconds verifying signs, basic arithmetic, and decimal point placements.

Essential Concept & Formula Summary:

To ensure total retention and swift revision under timed examination conditions, internalize the following consolidated principles:

• Axiomatic Rigor: Always verify that intermediate algebraic transformations satisfy underlying domain constraints. For square roots, ensure expressions under the radical are non-negative; for fractional expressions, ensure denominators are strictly non-zero.
• Standard Mathematical Notations: Use formal set-builder notations for solution sets, standard vector/matrix bracket conventions, and accurate geometric angle labels (e.g., using three-letter vertex angle descriptions like $\angle ABC$ rather than ambiguous single-letter labels).
• Systematic Verification: In algebraic solving (equations, factorisation, matrix equations, and proportions), always substitute final numerical solutions back into the original question to confirm that both Left-Hand and Right-Hand Sides evaluate to exact numerical equality.
• Graph Paper Guidelines: When working on coordinate geometry, reflections, histograms, and ogives, always indicate the origin clearly, write the axis names, write the chosen scale at the top right, and mark plotted points with neat small circles or crosses.