|x^2+4x|<12
To solve the inequality \( |x^2 + 4x| < 12 \), we start by rewriting it in a form that allows us to analyze the absolute value more easily.
The expression \( |x^2 + 4x| < 12 \) can be split into two separate inequalities:
- \( x^2 + 4x < 12 \)
- \( x^2 + 4x > -12 \)
Step 1: Solve \( x^2 + 4x < 12 \)
Rearranging gives us:
\[ x^2 + 4x - 12 < 0 \]
Next, we find the roots of the quadratic equation \( x^2 + 4x - 12 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 4, c = -12 \):
\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-12)}}{2 \cdot 1} \] \[ x = \frac{-4 \pm \sqrt{16 + 48}}{2} \] \[ x = \frac{-4 \pm \sqrt{64}}{2} \] \[ x = \frac{-4 \pm 8}{2} \]
This gives us the roots:
\[ x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-12}{2} = -6 \]
Now we find the intervals based on these roots: \( (-\infty, -6) \), \( (-6, 2) \), and \( (2, \infty) \).
Next, we test points within each interval to see where the inequality holds:
For \( x = -7 \) (in \( (-\infty, -6) \)): \[ (-7)^2 + 4(-7) - 12 = 49 - 28 - 12 = 9 \, (> 0) \]
For \( x = 0 \) (in \( (-6, 2) \)): \[ (0)^2 + 4(0) - 12 = -12 \, (< 0) \]
For \( x = 3 \) (in \( (2, \infty) \)): \[ (3)^2 + 4(3) - 12 = 9 + 12 - 12 = 9 \, (> 0) \]
Thus, \( x^2 + 4x < 12 \) holds in the interval \( (-6, 2) \).
Step 2: Solve \( x^2 + 4x > -12 \)
Rearranging gives us:
\[ x^2 + 4x + 12 > 0 \]
Next, we find the roots of the quadratic equation \( x^2 + 4x + 12 = 0 \). The discriminant will help us understand the roots:
\[ D = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 12 = 16 - 48 = -32 \]
Since the discriminant is negative, there are no real roots. The quadratic \( x^2 + 4x + 12 \) opens upwards (because \( a > 0 \)) and is always greater than zero for all \( x \).
Final Result
Combining the two results:
- \( x^2 + 4x < 12 \) gives \( x \in (-6, 2) \).
- \( x^2 + 4x > -12 \) gives all \( x \in \mathbb{R} \).
Thus, the solution to the inequality \( |x^2 + 4x| < 12 \) is:
\[ \boxed{(-6, 2)} \]
