Solve the following quadratic inequality: x2 – x – 4 ≤ 2

Explanation

Step 1: Move all terms to left side
x² – x – 4 ≤ 2
x² – x – 4 – 2 ≤ 0
x² – x – 6 ≤ 0

Step 2: Factorize the quadratic
x² – x – 6 = 0
Find factors of -6 that subtract to give -1: -3 and 2
(x – 3)(x + 2) ≤ 0

Step 3: Find critical points
Set each factor to zero:
x – 3 = 0 → x = 3
x + 2 = 0 → x = -2
Critical points: x = -2 and x = 3

Step 4: Test intervals
For x < -2 (try x = -3): (-)(-) = positive > 0 ✗
For -2 < x < 3 (try x = 0): (-)(+) = negative < 0 ✓
For x > 3 (try x = 4): (+)(+) = positive > 0 ✗

Step 5: Include boundary points
Since inequality is ≤ (not <), include x = -2 and x = 3
Therefore: -2 ≤ x ≤ 3