Find all solutions of the inequality
ln(x
2 − x − 12) ≤ ln 8 ,
expressing your answer in interval notation.

To solve the inequality \(\ln(x^2 - x - 12) \leq \ln 8\), we'll follow these steps:

### Step 1: Eliminate the logarithm
Since the natural logarithm is a monotonically increasing function, the inequality \(\ln(a) \leq \ln(b)\) implies that \(a \leq b\) as long as \(a > 0\) and \(b > 0\). So, we can rewrite the inequality as:
\[
x^2 - x - 12 \leq 8
\]

### Step 2: Solve the quadratic inequality
First, move the 8 to the left side:
\[
x^2 - x - 12 - 8 \leq 0
\]
\[
x^2 - x - 20 \leq 0
\]

Now, we solve the quadratic equation \(x^2 - x - 20 = 0\) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -1\), and \(c = -20\):
\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 80}}{2}
\]
\[
x = \frac{1 \pm \sqrt{81}}{2}
\]
\[
x = \frac{1 \pm 9}{2}
\]
\[
x = \frac{10}{2} = 5 \quad \text{or} \quad x = \frac{-8}{2} = -4
\]

So, the quadratic equation \(x^2 - x - 20 = 0\) has roots \(x = 5\) and \(x = -4\).

### Step 3: Determine the sign of the quadratic expression
The solutions divide the number line into three intervals:
1. \(x < -4\)
2. \(-4 \leq x \leq 5\)
3. \(x > 5\)

We test the sign of \(x^2 - x - 20\) in each interval.

- For \(x < -4\) (e.g., \(x = -5\)): \((-5)^2 - (-5) - 20 = 25 + 5 - 20 = 10\) (positive).
- For \(-4 < x < 5\) (e.g., \(x = 0\)): \(0^2 - 0 - 20 = -20\) (negative).
- For \(x > 5\) (e.g., \(x = 6\)): \(6^2 - 6 - 20 = 36 - 6 - 20 = 10\) (positive).

Thus, \(x^2 - x - 20 \leq 0\) holds in the interval \(-4 \leq x \leq 5\).

### Step 4: Ensure the argument of the logarithm is positive
We also need to ensure that the argument of the logarithm, \(x^2 - x - 12\), is positive because the logarithm is only defined for positive values.

Solve \(x^2 - x - 12 > 0\):
\[
x^2 - x - 12 = (x - 4)(x + 3) > 0
\]

This inequality is satisfied when \(x < -3\) or \(x > 4\).

### Step 5: Combine the intervals
The solution must satisfy both:
- \(-4 \leq x \leq 5\) from the quadratic inequality.
- \(x < -3\) or \(x > 4\) from the positivity condition.

The intersection of these intervals is:
\[
[-4, -3) \cup (4, 5]
\]

### Final Answer
The solution to the inequality \(\ln(x^2 - x - 12) \leq \ln 8\) in interval notation is:
\[
\boxed{[-4, -3) \cup (4, 5]}
\]

Would you like any details or have any questions? Here are 5 related questions for further practice:

1. Solve the inequality \(\ln(x^2 - 2x - 15) > \ln 3\).
2. Find the interval where \(\ln(2x^2 - 3x - 5) \geq \ln 4\).
3. Solve the equation \(\ln(x^2 - x - 12) = \ln 8\).
4. Determine the domain of the function \(f(x) = \ln(x^2 - 4x - 21)\).
5. Solve the inequality \(\ln(x^2 + 5x + 6) < \ln 1\).

**Tip:** Always consider the domain restrictions of logarithmic functions when solving inequalities.

Find all solutions of the inequality ln(x^2 − x − 12) ≤ ln 8, expressing your answer in interval notation.

Learn how to solve the inequality ln(x^2 − x − 12) ≤ ln 8 using properties of logarithms and the quadratic formula. This problem is ideal for students in Grades 10-12, covering key algebraic concepts and interval notation.

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