Solve quadratic inequalities by considering the positional relationship between the graph of y = ax² + bx + c and the x-axis.
Key Points for Solving
When D > 0 and the solutions are α < β (a > 0):
The solution of ax² + bx + c > 0: x < α or x > β
The solution of ax² + bx + c < 0: α < x < β
Example ①
Solve x² - 3x - 4 > 0.
Solution: Since (x + 1)(x - 4) > 0, x < -1 or x > 4
Solve x² - 3x - 4 > 0.
Solution: Since (x + 1)(x - 4) > 0, x < -1 or x > 4
Key Point
- First find the solutions of y = 0 (the x-intercepts).
- Above the x-axis → > 0; below the x-axis → < 0.
- When a < 0, pay attention to the direction of the inequality sign.
Practice Problems
- Solve x² - x - 6 ≤ 0.
- Solve x² - 4x + 4 > 0.
Solutions and Explanations
- Solution: -2 ≤ x ≤ 3. (x+2)(x-3) ≤ 0.
- Solution: All real numbers except x = 2. (x-2)² > 0 holds for all x except x = 2.