To solve the inequality \( |x^2 + x - 6| < 6 \), we will break it down step by step.
### Step 1: Remove the Absolute Value
We start by rewriting the inequality without the absolute value. This gives us two inequalities to solve:
\[
-6 < x^2 + x - 6 < 6
\]
### Step 2: Solve the First Inequality
We will first solve the left part of the compound inequality:
\[
x^2 + x - 6 > -6
\]
Adding 6 to both sides, we get:
\[
x^2 + x > 0
\]
### Step 3: Factor the Quadratic
Next, we can factor the quadratic expression:
\[
x(x + 1) > 0
\]
### Step 4: Find Critical Points
The critical points are found by setting the factors equal to zero:
\[
x = 0 \quad \text{and} \quad x + 1 = 0 \Rightarrow x = -1
\]
### Step 5: Test Intervals
Now we will test the intervals determined by the critical points \( -1 \) and \( 0 \):
1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \):
\[
(-2)(-2 + 1) = (-2)(-1) = 2 > 0 \quad \text{(True)}
\]
2. **Interval \( (-1, 0) \)**: Choose \( x = -0.5 \):
\[
(-0.5)(-0.5 + 1) = (-0.5)(0.5) = -0.25 < 0 \quad \text{(False)}
\]
3. **Interval \( (0, \infty) \)**: Choose \( x = 1 \):
\[
(1)(1 + 1) = (1)(2) = 2 > 0 \quad \text{(True)}
\]
Thus, the solution for \( x(x + 1) > 0 \) is:
\[
x \in (-\infty, -1) \cup (0, \infty)
\]
### Step 6: Solve the Second Inequality
Now we solve the right part of the compound inequality:
\[
x^2 + x - 6 < 6
\]
Subtracting 6 from both sides gives:
\[
x^2 + x - 12 < 0
\]
### Step 7: Factor the Quadratic
Next, we factor the quadratic expression:
\[
(x + 4)(x - 3) < 0
\]
### Step 8: Find Critical Points
The critical points are:
\[
x + 4 = 0 \Rightarrow x = -4 \quad \text{and} \quad x - 3 = 0 \Rightarrow x = 3
\]
### Step 9: Test Intervals
Now we will test the intervals determined by the critical points \( -4 \) and \( 3 \):
1. **Interval \( (-\infty, -4) \)**: Choose \( x = -5 \):
\[
(-5 + 4)(-5 - 3) = (-1)(-8) = 8 > 0 \quad \text{(False)}
\]
2. **Interval \( (-4, 3) \)**: Choose \( x = 0 \):
\[
(0 + 4)(0 - 3) = (4)(-3) = -12 < 0 \quad \text{(True)}
\]
3. **Interval \( (3, \infty) \)**: Choose \( x = 4 \):
\[
(4 + 4)(4 - 3) = (8)(1) = 8 > 0 \quad \text{(False)}
\]
Thus, the solution for \( (x + 4)(x - 3) < 0 \) is:
\[
x \in (-4, 3)
\]
### Step 10: Find the Intersection of Solutions
Now we need to find the intersection of the two solutions:
1. From \( x(x + 1) > 0 \): \( x \in (-\infty, -1) \cup (0, \infty) \)
2. From \( (x + 4)(x - 3) < 0 \): \( x \in (-4, 3) \)
The intersection is:
\[
x \in (-4, -1) \cup (0, 3)
\]
### Final Solution
Thus, the final solution to the inequality \( |x^2 + x - 6| < 6 \) is:
\[
\boxed{(-4, -1) \cup (0, 3)}
\]
