Quadratic Inequalities

Algebraic Mastery & Interval Analysis

Score 0 / 0

Accuracy 0%

Streak 0 🔥

Level 1 • Bounded Interval

Solve the 1-variable quadratic inequality:

x² - 5x + 6 < 0

Inequality Formula & Rules Cheat Sheet

1-Variable Inequalities (Critical Points & Sign Chart)

For ax^2 + bx + c = 0 with roots r₁ < r₂ (a > 0):

Bounded Interval (< 0 or ≤ 0): The solution set lies between the roots.
Inequality: r₁ < x < r₂ | Interval: (r₁, r₂)
Disjoint Intervals (> 0 or ≥ 0): The solution set lies outside the roots.
Inequality: x < r₁ or x > r₂ | Interval: (-∞, r₁) ∪ (r₂, ∞)
Endpoints Inclusion: Use brackets [ ] and ≤, ≥ for strict non-strict inequalities; parentheses ( ) and <, > for strict inequalities.

2-Variable Inequalities (y vs ax^2 + bx + c)

Boundary Line Type:
- ≤ or ≥ → Solid Curve (points on boundary are solutions)
- < or > → Dashed Curve (points on boundary are NOT solutions)
Shading / Half-Plane Condition:
- y > or y ≥ → Region Above parabola
- y < or y ≤ → Region Below parabola
Test Point Strategy: Plug a point like (0,0) into the inequality. If the statement is mathematically TRUE, (0,0) is inside the solution region.

Interval vs Inequality Notation

Inequality	Interval	Meaning
-2 < x < 5	(-2, 5)	Between -2 and 5 (exclusive)
-2 ≤ x ≤ 5	[-2, 5]	Between -2 and 5 (inclusive)
x < -2 or x > 5	(-∞, -2) ∪ (5, ∞)	Outside -2 and 5 (exclusive)