1. Introduction
Solving an inequality means finding all the values of the variable that make the inequality true. Unlike equations, inequalities usually have a range of values instead of a single answer.
Example: Solving x > 3 gives infinitely many solutions: 4, 5, 10, 100, 3.5, etc.
2. Important Rules for Solving Inequalities
Solving inequalities is similar to solving equations, but with one major difference: the direction of the inequality may change.
2.1. Rule 1: Addition Rule
You can add the same number to both sides of an inequality.
Example: x − 5 > 2 → x > 7
2.2. Rule 2: Subtraction Rule
You can subtract the same number from both sides.
Example: x + 3 ≥ 9 → x ≥ 6
2.3. Rule 3: Multiplication Rule
You can multiply both sides by a positive number without changing the sign.
Example: x/4 > 2 → x > 8
2.4. Rule 4: Division Rule
You can divide both sides by a positive number without changing the sign.
Example: 3x ≥ 15 → x ≥ 5
3. Most Important Rule: Sign Change Rule
If you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed.
Example:
- −2x > 8 → divide by −2 → x < −4
Why? Because negative multiplication flips the direction of the comparison.
4. Solving Linear Inequalities in One Variable
To solve a linear inequality:
- Simplify both sides.
- Collect variable terms on one side.
- Use rules of addition/subtraction.
- Divide by coefficient of the variable (apply sign rule if needed).
4.1. Example 1
Solve: 3x − 7 < 11
3x < 18 → x < 6
4.2. Example 2
Solve: 5 − 2x ≥ 1
−2x ≥ −4 → divide by −2 → x ≤ 2
4.3. Example 3
Solve: 2(x − 1) > 3x + 4
2x − 2 > 3x + 4 → −x > 6 → x < −6
5. Describing Solution Sets
After solving, we often write the solution using special notations.
5.1. 1. Set-Builder Form
Example: x > 3 → { x ∈ R : x > 3 }
5.2. 2. Interval Notation
(3, ∞) → all real numbers greater than 3
[a, b] means a and b included.
(a, b) means endpoints excluded.
6. Examples (Easy to Medium)
- Solve: x + 5 ≤ 12 → x ≤ 7
- Solve: −3x > 9 → x < −3
- Solve: 4 − (x/2) ≥ 1 → −x/2 ≥ −3 → x ≤ 6
7. Common Mistakes
- Forgetting to reverse the inequality when multiplying/dividing by a negative number.
- Changing the inequality unnecessarily when adding/subtracting.
- Mixing up open and closed intervals.
- Thinking the solution is a single number instead of a range.
8. Quick Practice
Solve the following:
- 7 − 2x > 3
- 3x + 1 ≤ 10
- −5x ≥ 20
- 2(x + 3) < 5x − 2
9. Summary
- Solving inequalities gives a range of valid values.
- All rules of equations apply except when multiplying or dividing by a negative number.
- Solutions can be written using set-builder or interval notation.
- Always check whether the boundary is included (≤ or ≥) or not (< or >).