CBSE Class 11 Maths: Linear Inequalities Notes

Important Glossary & Definitions

Inequality

A mathematical statement that uses inequality relation symbols ($<$, $>$, $\le$, $\ge$) to compare the relative size of two values or algebraic expressions.

Strict Inequality

An inequality that contains only the relations '$<$' (less than) or '$>$' (greater than), excluding the possibility of equality.

Slack (Semi-strict) Inequality

An inequality that contains the relations '$\le$' (less than or equal to) or '$\ge$' (greater than or equal to), which includes potential equality.

Solution of an Inequality

Any real value or set of values of the variable(s) that satisfies the inequality and makes the statement mathematically true.

Solution Set

The complete set of all possible real numbers or values that satisfy a given mathematical inequality.

Interval Notation

A mathematical system for representing subsets of real numbers using brackets, where open brackets $()$ denote strict bounds and closed brackets $[]$ denote inclusive bounds.

Algebraic Properties and Sign Reversal Principle

To master linear inequalities, you must understand how algebraic operations differ from standard equations. Adding or subtracting the same real number on both sides of an inequality does not affect the inequality sign. Similarly, multiplying or dividing both sides by a positive real number preserves the direction of the inequality.

However, the most critical rule—and the most common source of marks deduction in exams—is the sign reversal principle. When both sides of an inequality are multiplied or divided by a negative real number, the inequality sign is reversed. For example, if $-3x < 9$, dividing by $-3$ transforms the relation into $x > -3$. Additionally, taking reciprocals of positive quantities reverses the inequality sign (if $a > b > 0$, then $\frac{1}{a} < \frac{1}{b}$).

Step-by-Step Guide: Solving and Representing 1D Inequalities

1. Simplify both sides —
2. Isolate the variable terms —
3. Apply multiplication or division rules —
4. Write in interval notation —
5. Represent on number line —

Comparison: Strict vs. Slack Inequalities

Must-Remember Rules for Linear Inequalities

• Equal numbers can be added to (or subtracted from) both sides of an inequality without affecting its sign direction.
• Both sides of an inequality can be multiplied (or divided) by the same positive number without changing the sign.
• Multiplying or dividing both sides of an inequality by a negative number ALWAYS reverses the direction of the inequality sign.
• While writing solution sets in interval notation, infinity ($\infty$) and negative infinity ($-\infty$) must always be enclosed with open parentheses, i.e., $( \dots, \infty)$.
• Do not cross-multiply terms containing variables if their signs are unknown, as variables can represent negative values.
• For graphical representation of linear inequalities in two variables ($ax + by \le c$), use the origin test $(0,0)$ to quickly identify the correct half-plane region.
• If the origin test yields a true statement, the shaded region contains the origin; if it is false, the shaded region lies on the opposite side of the boundary line.

Worked Examples for CBSE Quick Revision

• Example 1: Algebraic Solution of 1D Inequality
  Q: Solve $4x + 3 < 6x + 7$ for real values of $x$.
  A: Step 1: Subtract $4x$ from both sides: $3 < 2x + 7$.
  Step 2: Subtract $7$ from both sides: $-4 < 2x$.
  Step 3: Divide by positive $2$ (no sign reversal): $-2 < x$ or $x > -2$.
  Solution Set in Interval Notation: $(-2, \infty)$.

• Example 2: Inequality involving fractions and negative coefficient
  Q: Solve $\frac{5-2x}{3} \le \frac{x}{6} - 5$ for real numbers.
  A: Step 1: Multiply the entire inequality by $6$ (LCM) to clear denominators: $2(5-2x) \le x - 30$.
  Step 2: Expand and simplify: $10 - 4x \le x - 30$.
  Step 3: Rearrange terms: $10 + 30 \le x + 4x \implies 40 \le 5x$.
  Step 4: Divide by $5$: $8 \le x$ or $x \ge 8$.
  Solution Set in Interval Notation: $[8, \infty)$.

CBSE Exam Warning: Common Trap Alert!

⚠️ The Negative Coefficient Trap: In exams, the most frequent mistake made by CBSE Class 11 students is forgetting to reverse the inequality direction when multiplying or dividing by a negative number. For instance, transforming $-x > 5$ into $x > -5$ is mathematically incorrect and will result in a zero mark for that step. The correct reduction is $x < -5$. Always double-check your final intervals by choosing a random value from your solution set and substituting it back into the original inequality.

Quick Knowledge Check

• Explain when the inequality symbol is reversed with an example.
  The inequality symbol is reversed when both sides are multiplied or divided by a negative number, or when taking the reciprocal of expressions of the same sign. Example: $-2x < 10 \implies x > -5$.

• What does a closed bracket $[a, b]$ represent in interval notation compared to open brackets $(a, b)$?
  A closed bracket $[a, b]$ includes the endpoints $a$ and $b$ (slack bounds, $a \le x \le b$), whereas open brackets $(a, b)$ exclude them (strict bounds, $a < x < b$).

• How is the graphical region of $x \ge -3$ plotted on a one-dimensional number line?
  Place a solid/shaded circle at $-3$ on the number line and shade/draw a thick arrow extending to the right side towards positive infinity.

• Can we cross-multiply variable expressions in inequalities, e.g., $\frac{1}{x} < 2 \implies 1 < 2x$?
  No, cross-multiplication is not safe unless you are sure of the sign of the variable expression. If $x$ is negative, multiplying by $x$ would reverse the inequality sign. Instead, bring all terms to one side and solve.

Frequently Asked Questions