An equation says two quantities are identical ($x = 5$). An inequality says one quantity is greater than, less than, or bounded by another ($x < 5$ or $x \ge 5$).
Strict Inequalities ($<, >$)
The boundary value itself is strictly excluded.
Examples: $ax + b < 0$, $ax + b > 0$.
Number line representation uses a hollow circle ($\circ$). 2D graphs use a dashed boundary line.
Slack / Non-Strict Inequalities ($\le, \ge$)
The boundary value is included ("less than or equal to").
Examples: $ax + b \le 0$, $ax + b \ge 0$.
Number line representation uses a solid dot ($ullet$). 2D graphs use a solid boundary line.
2. The 3 Golden Rules of Inequality Transformations
Rule 1: Addition & Subtraction
Equal numbers may be added to (or subtracted from) both sides of an inequality without changing the direction of the inequality sign:
$$a < b \implies a + c < b + c \quad \text{and} \quad a - c < b - c$$
Rule 2: Multiplication/Division by a Positive Number
Both sides of an inequality can be multiplied (or divided) by the same positive number ($c > 0$) without changing the direction of the inequality sign:
$$a < b \implies ac < bc \quad \text{and} \quad \frac{a}{c} < \frac{b}{c} \quad (\text{when } c > 0)$$
Rule 3 (THE MOST IMPORTANT RULE): Multiplying/Dividing by a Negative Number
When both sides of an inequality are multiplied (or divided) by the same negative number ($c < 0$), the inequality sign MUST BE REVERSED!
$$a < b \implies ac > bc \quad \text{and} \quad \frac{a}{c} > \frac{b}{c} \quad (\text{when } c < 0)$$
Intuition: We know $2 < 5$. If we multiply both sides by $-1$, $-2$ is greater than $-5$ on the number line! Hence $-2 > -5$.
3. Interval Notation & Number Line
Inequality
Interval Notation
Type
Graphic Meaning
$a < x < b$
$(a, b)$
Open Interval
All real numbers between $a$ and $b$; endpoints excluded.
$a \le x \le b$
$[a, b]$
Closed Interval
All real numbers between $a$ and $b$; both endpoints included.
$a \le x < b$
$[a, b)$
Semi-closed / Semi-open
$a$ is included (square bracket), $b$ is excluded (round bracket).
$x > a$
$(a, \infty)$
Unbounded Above
Ray extending infinitely to the right from $a$. $\infty$ always has round bracket!
$x \le b$
$(-\infty, b]$
Unbounded Below
Ray extending infinitely to the left up to $b$ inclusive.
Dependence on Domain: $\mathbb{N}$ vs $\mathbb{Z}$ vs $\mathbb{R}$
Consider the inequality: $24x < 100 \implies x < \frac{100}{24} = \frac{25}{6} = 4.166...$
If $x \in \mathbb{N}$ (Natural numbers): Solution set is $\{1, 2, 3, 4\}$ (discrete elements in curly braces).
If $x \in \mathbb{Z}$ (Integers): Solution set is $\{..., -2, -1, 0, 1, 2, 3, 4\}$.
If $x \in \mathbb{R}$ (Real numbers): Solution set is an infinite continuum: $(-\infty, 25/6)$ in interval notation!
4. Solving Systems of Linear Inequalities in One Variable
Step-by-Step Method
Solve each inequality independently for $x$.
Plot the solution of each inequality on the same real number line.
The overall solution is the intersection (overlapping region) of all individual solutions.
Write the common overlapping region in interval notation.
5. Graphical Solution in Two Variables (Half-Planes)
Step 1: Draw Boundary Line
Replace inequality sign with equality: $ax + by = c$. Plot by finding $x$-intercept (set $y=0$) and $y$-intercept (set $x=0$).
Draw solid line for $\le, \ge$; draw dashed line for $<, >$.
Step 2: Test Point Method
Pick an easy point not on the line, usually the origin $(0, 0)$. Substitute $(0, 0)$ into the inequality.
If true: shade the half-plane containing $(0,0)$. If false: shade the opposite half-plane.
6. Solving Classic Word Problems
1. Examination Average Problem
If a student's marks in two tests are $m_1, m_2$, and they need an average of at least $A$ across 3 tests:
$$\frac{m_1 + m_2 + x}{3} \ge A \implies x \ge 3A - (m_1 + m_2)$$
2. Mixture Problems
When mixing $x$ liters of $c_1\%$ solution with $y$ liters of $c_2\%$ solution, the total amount of solute is:
$$\frac{c_1}{100}x + \frac{c_2}{100}y$$
Divide by total volume $(x + y)$ to find the resulting concentration.
7. Worked Examples — Easy to Hard
Example 1: Single Inequality with Negative CoefficientEasy
Solve $-12x > 30$, when: (i) $x$ is a natural number, (ii) $x$ is an integer, (iii) $x$ is a real number.
Step 1: Divide by $-12$ and REVERSE sign:
$$x < \frac{30}{-12} = -\frac{5}{2} = -2.5$$
(i) $x \in \mathbb{N}$: Natural numbers are $\{1, 2, 3, ...\}$. None are $< -2.5$. Solution set is $\phi$ (empty set).
(ii) $x \in \mathbb{Z}$: Integers $< -2.5$ are $\{..., -5, -4, -3\}$.
(iii) $x \in \mathbb{R}$: Interval is $(-\infty, -5/2)$.
Example 2: Fractions on Both SidesMedium
Solve the inequality for real $x$:
$$\frac{3(x - 2)}{5} \le \frac{5(2 - x)}{3}$$
Step 1: Multiply both sides by LCM $= 15$ (positive, so sign does not change):
$$3 \times 3(x - 2) \le 5 \times 5(2 - x)$$
$$9(x - 2) \le 25(2 - x)$$
Solve: $-5 \le \frac{5 - 3x}{2} \le 8$ for real $x$.
Step 1: Multiply all sides by $2$:
$$-10 \le 5 - 3x \le 16$$
Step 2: Subtract $5$ from all parts:
$$-15 \le -3x \le 11$$
Step 3: Divide by $-3$ and REVERSE all inequality signs:
$$\frac{-15}{-3} \ge x \ge \frac{11}{-3} \implies 5 \ge x \ge -\frac{11}{3}$$
Rewriting in standard increasing order:
$$-\frac{11}{3} \le x \le 5$$
$$\mathbf{\text{Solution: } \left[-\frac{11}{3}, 5\right]}$$
Example 5: Word Problem — Acid MixtureHard
A manufacturer has 600 liters of a $12\%$ solution of acid. How many liters of a $30\%$ acid solution must be added to it so that acid content in the resulting mixture will be more than $15\%$ but less than $18\%$?
Step 1: Define variables:
Let $x$ liters of $30\%$ acid solution be added.
Total volume of mixture $= 600 + x$ liters.
Total acid content $= 12\% \text{ of } 600 + 30\% \text{ of } x = \frac{12}{100}(600) + \frac{30}{100}(x) = 72 + 0.3x$.
Step 2: Set up inequalities:
$$15\% \text{ of } (600 + x) < 72 + 0.3x < 18\% \text{ of } (600 + x)$$
$$0.15(600 + x) < 72 + 0.3x < 0.18(600 + x)$$
Step 4: Solve second inequality:
$$72 + 0.3x < 108 + 0.18x \implies 0.12x < 36 \implies x < 300$$
Conclusion: Between $120$ liters and $300$ liters of $30\%$ acid solution must be added.
8. NCERT Key Exercise Questions Solved
NCERT Exercise 5.1 — Question 21
Q: Ravi obtained 70 and 75 marks in first two unit tests. Find the minimum marks he should get in the third test to have an average of at least 60 marks.
View Full Solution
Let $x$ be the marks in the third test.
$$\frac{70 + 75 + x}{3} \ge 60$$
$$145 + x \ge 180 \implies x \ge 180 - 145 = 35$$
Minimum marks required = 35.
NCERT Exercise 5.1 — Question 23
Q: Find all pairs of consecutive odd positive integers, both of which are smaller than 10, such that their sum is more than 11.
View Full Solution
Let the smaller odd positive integer be $x$. The next consecutive odd integer is $x + 2$.
Condition 1: Both smaller than 10: $x + 2 < 10 \implies x < 8$.
Condition 2: Sum more than 11: $x + (x + 2) > 11 \implies 2x + 2 > 11 \implies 2x > 9 \implies x > 4.5$.
Since $x$ is an odd positive integer and $4.5 < x < 8$, the possible values for $x$ are $5$ and $7$.
Pairs $(x, x+2)$ are: $(5, 7)$ and $(7, 9)$.
9. Common Exam Traps & Pitfalls
Trap 1: Cross-Multiplying Variables
In an inequality like $\frac{1}{x} < 1$, NEVER cross-multiply $x$ to get $1 < x$! Because if $x$ is negative, multiplying by $x$ would flip the sign. Always bring everything to one side: $\frac{1}{x} - 1 < 0 \implies \frac{1 - x}{x} < 0$, and use sign charts.
Trap 2: Forgetting to Reverse Sign on Negative Division
The single most common mistake in board exams: Writing $-2x \le 8 \implies x \le -4$. IT IS WRONG! Dividing by $-2$ reverses the sign to $x \ge -4$.
Trap 3: Confusing Natural Numbers with Real Numbers
If the question says "where $x$ is an integer", do NOT write $(-\infty, 3)$ as your final answer. Write $\{..., -1, 0, 1, 2\}$. Bracket notation like $(a, b)$ is strictly reserved for real numbers.
10. Quick Self-Check Quiz
Q1: If $-3x + 17 < -13$, what is the range of real $x$?
Answer
$-3x < -30 \implies x > 10$. In interval notation: $(10, \infty)$.
Q2: How many positive integers satisfy $4x + 3 < 6x + 7$?
Answer
$4x - 6x < 7 - 3 \implies -2x < 4 \implies x > -2$. Since all positive integers are $> -2$, there are infinitely many solutions ($x \in \{1, 2, 3, ...\}$).
Q3: What is the intersection of $[-3, 7)$ and $(2, 10]$?
Answer
$(2, 7)$. (2 is excluded because it's excluded in $(2, 10]$, 7 is excluded because it's excluded in $[-3, 7)$).
Q4: In a graph of $2x - 3y > 6$, is the boundary line solid or dashed?
Answer
Dashed, because the inequality is strict ($>$).
Q5: Does the point $(0, 0)$ satisfy $3x - 5y \le 15$?
Answer
Yes, $3(0) - 5(0) = 0 \le 15$ is true, so $(0, 0)$ lies in the solution half-plane.