Chapter in a nutshell: A linear inequation uses <, >, ≤, ≥ instead of "=". You solve it like an equation, but with one crucial rule: multiplying or dividing by a negative number reverses the inequality sign. The answer is a solution set (depending on the replacement set N, W, Z or R) shown on a number line.
1. Key Ideas & Terms
- Inequation: an open statement with <, >, ≤, ≥.
- Replacement (domain) set: the set from which solutions are taken — N (naturals), W (wholes), Z/I (integers) or R (reals).
- Solution set: the values from the replacement set that satisfy the inequation.
- On a number line: ● filled dot = value included (≤, ≥); ○ open dot = excluded (<, >); for R, shade the ray.
2. Rules for Solving (the formula set)
- Add or subtract the same number on both sides → sign unchanged.
- Multiply or divide by a positive number → sign unchanged.
- Multiply or divide by a NEGATIVE number → reverse the inequality sign ( < ↔ > , ≤ ↔ ≥ ).
- Transposition: a term changes sign when moved across; the inequality sign stays (this is just rule 1).
3. Method (steps)
- Simplify both sides (remove brackets/fractions — multiply through by the LCM, watching sign rule 3 if the multiplier is negative).
- Collect the variable on one side, constants on the other.
- Make the coefficient of x = 1 (divide; reverse sign if dividing by a negative).
- Pick the solutions from the replacement set and show them on the number line.
4. Worked Example (one, for the method)
Q. Solve $3 - 2x \ge 7$, x ∈ Z, and show on a number line. Solution: $-2x \ge 4 \Rightarrow x \le -2$ (sign reversed on dividing by −2). Solution set = {…, −4, −3, −2}; on the line, a filled dot at −2 with the ray to the left.5. Prerequisite Ideas (earlier classes)
- Number systems: N ⊂ W ⊂ Z ⊂ Q ⊂ R.
- Solving linear equations and transposition (same skills, plus the sign-reversal rule).
6. Common Mistakes to Avoid
- Forgetting to reverse the sign when multiplying/dividing by a negative.
- Ignoring the replacement set (e.g. listing negatives when the set is N).
- Using the wrong dot on the number line (open vs filled).
- Reversing the sign for addition/subtraction (only ×/÷ by a negative reverses it).
7. Likely Exam Questions (with crisp answers)
- When does the inequality sign reverse? → When you multiply or divide both sides by a negative number.
- What is a replacement set? → The set from which the solution values are chosen.
- Show ≤ on a number line. → A filled dot at the value with a ray in the direction of the solutions.
- Does transposing a term change the inequality sign? → No.
- Solve $x + 5 < 9$, x ∈ N. → x < 4 → {1, 2, 3}.
- Solve $-3x \le 9$. → x ≥ −3 (sign reversed).
- Difference between an open and filled dot? → Open = value excluded (<,>); filled = value included (≤,≥).
- What sets do N, W, Z, R stand for? → Natural, whole, integers, real numbers.
Formulas — Current & Prerequisite
Current
- Add/subtract same number to both sides: inequality unchanged.
- Multiply/divide both sides by a negative number ⇒ reverse the sign.
- Solution sets over $x\in\mathbb{N},\mathbb{W},\mathbb{Z},\mathbb{R}$; represent on the number line (open ∘ for strict, closed • for ≤,≥).
Prerequisite
- Transposition and solving linear equations; properties of inequalities; integers/real number line.