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:
- \( (-\infty, -6) \)
- \( (-6, 5) \)
- \( (5, \infty) \)
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.
