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4.4 Quadratic Inequalities

Ethiopian Grade 9 Mathematics - Unit 4

1. Learning Objectives

2. Solving Methods

Step 1: Factor the quadratic expression (if possible).
Step 2: Find critical points by setting each factor to zero.
Step 3: Create a sign chart or test values in intervals.
Step 4: Select intervals that satisfy the inequality (> 0 for positive, < 0 for negative).
✍️ Practice Exercises (20 Questions)

Click the button below each question to reveal the explanation.

1. Solve: (x - 2)(x + 3) > 0
Critical points: 2, -3. Test intervals: (-∞, -3) is (+), (-3, 2) is (-), (2, ∞) is (+). We want > 0. Ans: (-∞, -3) ∪ (2, ∞)
2. Solve: x² - 9 < 0
Factor: (x-3)(x+3) < 0. Critical points: 3, -3. The middle interval is negative. Ans: (-3, 3)
3. Solve: x² ≥ 16
x² - 16 ≥ 0 → (x-4)(x+4) ≥ 0. Outside intervals are positive. Ans: (-∞, -4] ∪
8. Solve: x² - 4x + 4 > 0
Factor: (x-2)² > 0. True for all x except when x=2. Ans: x ≠ 2
9. Solve: -x² + 4 > 0
Multiply by -1 and flip: x² - 4 < 0 → (x-2)(x+2) < 0. Ans: (-2, 2)
10. Solve: x² + 4x + 3 < 0
Factor: (x+1)(x+3) < 0. Critical points: -1, -3. Ans: (-3, -1)
11. Solve: x² ≤ 0
x² is never negative, but equals 0 at x=0. Ans: x = 0
12. Solve: (x + 5)² ≥ 0
A squared number is always greater than or equal to zero. Ans: All Real Numbers
13. Solve: x² - x - 12 ≤ 0
Factor: (x-4)(x+3) ≤ 0. Ans: [-3, 4]
14. If the discriminant D < 0 and a > 0, solve ax² + bx + c < 0.
The graph is entirely above the x-axis. Ans: No Solution
15. Solve: 2x² - 8 > 0
Divide by 2: x² - 4 > 0 → (x-2)(x+2) > 0. Ans: (-∞, -2) ∪ (2, ∞)
16. Solve: x² + 6x + 8 ≥ 0
Factor: (x+2)(x+4) ≥ 0. Critical points: -2, -4. Ans: (-∞, -4] ∪
20. Solve: x² - 2x + 1 < 0
Factor: (x-1)² < 0. A square cannot be negative. Ans: No Solution