X²+x-30<0

To solve the inequality \( x^2 + x - 30 < 0 \), we first need to find the roots of the corresponding equation \( x^2 + x - 30 = 0 \). ### Step 1: Find the roots We can factor the quadratic: \[ x^2 + x - 30 = (x - 5)(x + 6) = 0 \] This gives us the roots: \[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \] \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \] ### Step 2: Determine intervals The roots \( x = -6 \) and \( x = 5 \) divide the number line into three intervals: 1. \( (-\infty, -6) \) 2. \( (-6, 5) \) 3. \( (5, \infty) \) ### Step 3: Test each interval Now we will test a point from each interval to see where the inequality \( (x - 5)(x + 6) < 0 \) holds true. - **Interval 1: \( (-\infty, -6) \)** - Test \( x = -7 \): \[ (-7 - 5)(-7 + 6) = (-12)(-1) = 12 \quad (> 0) \quad \text{(not in the solution)} \] - **Interval 2: \( (-6, 5) \)** - Test \( x = 0 \): \[ (0 - 5)(0 + 6) = (-5)(6) = -30 \quad (< 0) \quad \text{(in the solution)} \] - **Interval 3: \( (5, \infty) \)** - Test \( x = 6 \): \[ (6 - 5)(6 + 6) = (1)(12) = 12 \quad (> 0) \quad \text{(not in the solution)} \] ### Step 4: Conclusion The inequality \( x^2 + x - 30 < 0 \) holds true for the interval: \[ \boxed{(-6, 5)} \] This means that for any \( x \) in the interval \( (-6, 5) \), the quadratic expression will be negative.