Equations & Inequalities
Always verify by substituting back in. Flip the sign only when dividing by a negative.
A. Linear Equations
Linear Equation
An equation where the variable has power 1 only. Solve by doing the same operation on both sides until the variable is alone.
RuleWhatever you do to one side — add, subtract, multiply, divide — you must do the exact same thing to the other side.
Example 1 — Solve 3x + 7 = 22
1Subtract 7 from both sides: 3x = 22 − 7 = 15
2Divide both sides by 3: x = 15 ÷ 3 = 5
✓Check: 3(5)+7 = 22 ✓
x = 5
Example 2 — Solve 5x − 3 = 2x + 9
1Move x to left: 5x − 2x = 9 + 3
23x = 12 → x = 4
x = 4 — verify: 5(4)−3=17, 2(4)+9=17 ✓
Example 3 — Word problem: A number doubled plus 9 = 37
1Let the number = x → equation: 2x + 9 = 37
22x = 28 → x = 14
The number is 14
B. Quadratic Equations
Quadratic Equation
An equation with x² in it. Form: ax² + bx + c = 0. Can have two, one, or zero solutions.
Examplex² + 5x + 6 = 0 is quadratic. The highest power is 2.
Discriminant (D)
D = b² − 4ac. Tells you how many solutions exist before you even solve.
ResultD > 0 → two solutions D = 0 → one solution D < 0 → no real solution
Quadratic Formula
x = [−b ± √(b² − 4ac)] ÷ 2a
Example 1 — Solve x² + 5x + 6 = 0 by factoring
1Need: product = 6 and sum = 5 → 2 and 3
2(x + 2)(x + 3) = 0
3x + 2 = 0 → x = −2 OR x + 3 = 0 → x = −3
x = −2 or x = −3
Example 2 — Solve 2x² − 5x + 2 = 0 using formula
1a=2, b=−5, c=2
2D = (−5)² − 4(2)(2) = 25 − 16 = 9
3x = (5 ± √9) ÷ 4 = (5 ± 3) ÷ 4
4x = (5+3)÷4 = 2 OR x = (5−3)÷4 = 0.5
x = 2 or x = 0.5
C. Simultaneous Equations
Elimination Method
Add or subtract the two equations to remove one variable, then solve for the other.
Best whenBoth equations have the same coefficient for one variable.
Substitution Method
Solve one equation for one variable, then plug that into the other equation.
Best whenOne equation is already in the form y = ... or x = ...
Elimination — x + y = 10 and x − y = 4
1Add both equations: 2x = 14 → x = 7
2Sub into eq1: 7 + y = 10 → y = 3
x = 7, y = 3 — verify: 7+3=10 ✓ 7−3=4 ✓
Substitution — y = 2x − 1 and 3x + y = 14
1Replace y with (2x−1): 3x + (2x−1) = 14
25x − 1 = 14 → 5x = 15 → x = 3
3y = 2(3) − 1 = 5
x = 3, y = 5
D. Inequalities
> (greater than) and < (less than)
Strict — does not include the boundary value.
Examplex > 5 means x can be 6, 7, 8 … but NOT 5 itself.
≥ (greater than or equal) and ≤ (less than or equal)
Includes the boundary value.
Examplex ≥ 5 means x can be 5, 6, 7, 8 … (5 is included).
Number Lines — Visualising Inequalities
The Flip Rule
When you multiply or divide both sides by a negative number, the inequality sign must flip direction.
Example−2x > 8 → divide by −2 → x < −4 (sign flipped!)
Example 1 — Solve 3x − 4 > 11
13x > 11 + 4 = 15
2x > 15 ÷ 3 = 5 (dividing by positive 3, no flip)
x > 5
Example 2 — Solve −4x ≥ 20 (FLIP!)
1Divide both sides by −4 → must FLIP the sign
2x ≤ 20 ÷ (−4) → x ≤ −5
x ≤ −5 (sign changed from ≥ to ≤)
⚠ The flip rule: Only flip when multiplying or dividing by a negative. Adding/subtracting does NOT flip.
⚡ MCQ Tip: Always verify your linear equation answer by substituting back in. For quadratics, check the discriminant first — if D < 0 you can immediately say "no real roots" and skip solving. For inequalities, flip the sign only when dividing by a negative.
Quick MCQ Revision
| Rule | Remember |
|---|---|
Quad formula | [−b ± √(b²−4ac)] ÷ 2a |
Discriminant | D>0 → 2 roots · D=0 → 1 root · D<0 → no real roots |
Flip rule | Multiply/divide by NEGATIVE → flip the sign |
| Elimination | Add/subtract equations to cancel one variable |
| Substitution | Solve one eq for one variable, plug into other |