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We are given the following matrix inequality to solve:

\[
\left| \begin{matrix} 
x & x \\
x & -4 
\end{matrix} \right| > 3
\]

### Step 1: Find the determinant of the matrix

The determinant of a 2x2 matrix \(\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]\) is calculated as:

\[
\text{determinant} = ad - bc
\]

For the given matrix:

\[
\left[ \begin{matrix} 
x & x \\
x & -4 
\end{matrix} \right]
\]

The determinant is:

\[
\text{det} = (x)(-4) - (x)(x) = -4x - x^2 = -x^2 - 4x
\]

### Step 2: Solve the inequality

We now have the inequality:

\[
|-x^2 - 4x| > 3
\]

This absolute value inequality can be rewritten as two separate inequalities:

1. \( -x^2 - 4x > 3 \)
2. \( -x^2 - 4x < -3 \)

Let's solve each of these cases.

---

### Case 1: \( -x^2 - 4x > 3 \)

Rearrange the inequality:

\[
-x^2 - 4x - 3 > 0
\]

Multiply through by \(-1\) (remembering to flip the inequality):

\[
x^2 + 4x + 3 < 0
\]

Factor the quadratic:

\[
(x + 1)(x + 3) < 0
\]

Using the sign chart, this inequality holds for \( -3 < x < -1 \).

---

### Case 2: \( -x^2 - 4x < -3 \)

Rearrange the inequality:

\[
-x^2 - 4x + 3 < 0
\]

Multiply through by \(-1\) (again, flipping the inequality):

\[
x^2 + 4x - 3 > 0
\]

Solve the quadratic equation \(x^2 + 4x - 3 = 0\) using the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{16 + 12}}{2} = \frac{-4 \pm \sqrt{28}}{2} = \frac{-4 \pm 2\sqrt{7}}{2}
\]

Thus, the solutions are:

\[
x = -2 \pm \sqrt{7}
\]

So, \(x^2 + 4x - 3 > 0\) holds outside the roots, i.e., for \(x < -2 - \sqrt{7}\) or \(x > -2 + \sqrt{7}\).

---

### Final solution

Combining the solutions from both cases, we have:

\[
x \in (-3, -1)
\]

Would you like more details on any of the steps? Here are 5 related questions you might explore:

1. How do you solve absolute value inequalities in general?
2. What is the determinant of a matrix, and why is it important in solving matrix-related inequalities?
3. Can you show the full process of factoring quadratic equations?
4. How does the quadratic formula work for solving inequalities?
5. What are some methods for solving inequalities involving absolute values?

**Tip:** Always check the sign of coefficients carefully when manipulating inequalities, as it impacts the direction of the inequality sign.

Решите неравенство |x x| > 3

Learn how to solve the matrix inequality |x x| > 3, including finding the determinant and solving the quadratic and absolute value inequalities step by step. Suitable for high school students in Grades 10-11.

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