Algebra (Identities and Inequations) — Class 7 Maths (Samacheer Kalvi)
TN State Board (Samacheer Kalvi) Class 7 Mathematics, Term 3 — Chapter 3. Algebraic identities and linear inequalities.
1. About this chapter
This chapter covers algebraic identities and their expansions, the expansion of (x + a)(x + b), and linear inequations — solving them and showing them on a number line.
2. Algebraic identities
An identity is an equation true for all values of the variables. The standard identities are:
| Identity | Expansion |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| (a + b)(a − b) | a² − b² |
| (x + a)(x + b) | x² + (a + b)x + ab |
3. Using identities
- They give a quick way to expand or compute. Example: 102² = (100 + 2)² = 100² + 2·100·2 + 2² = 10000 + 400 + 4 = 10404.
- Example: 99 × 101 = (100 − 1)(100 + 1) = 100² − 1² = 9999.
4. Linear inequations
- An inequation uses <, >, ≤ or ≥ instead of "=" (e.g. x + 3 > 7).
- Solve like an equation by doing the same operation on both sides, but if you multiply or divide by a negative number, reverse the inequality sign.
- Example: x + 3 > 7 → x > 4. On the number line, show all points to the right of 4 (open circle at 4).
5. Worked examples
Example 1. Expand (x + 5)². x² + 2·x·5 + 5² = x² + 10x + 25.
Example 2. Expand (y − 3)(y − 3) using an identity. (y − 3)² = y² − 6y + 9.
Example 3. Solve 2x − 1 < 9. 2x < 10 → x < 5.
6. Exercises (Samacheer Kalvi)
- Expand: (a) (a + 7)² (b) (m − 4)².
- Expand using an identity: (x + 6)(x − 6).
- Find 103² using an identity.
- Expand (x + 2)(x + 5).
- Solve and show on a number line: (a) x − 2 ≥ 3 (b) 3x ≤ 12.
7. Common mistakes
- Mistake: Writing (a + b)² = a² + b². Fix: (a + b)² = a² + 2ab + b² — don't forget the middle term 2ab.
- Mistake: Forgetting to reverse the sign when dividing by a negative. Fix: Multiplying/dividing an inequation by a negative flips < to >.
- Mistake: Mixing up (a − b)² and (a + b)(a − b). Fix: (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b².
8. Quick revision
- Term 3 · Ch 3 · identities and inequations.
- (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b²; (x + a)(x + b) = x² + (a + b)x + ab.
- Use identities to expand and compute quickly (e.g. 99 × 101 = 9999).
- Solve inequations like equations; reverse the sign when multiplying/dividing by a negative; show the solution on a number line.
