https://file.aiquickdraw.com/mathbot/file/1727119106303igpb68nw.jpg

The inequality in the image involves solving a 2x2 determinant inequality:

\[
\left| \begin{matrix} -6 & 9 \\ x & 5 \end{matrix} \right| > 0
\]

To solve this, we first compute the determinant of the 2x2 matrix. The determinant \(D\) of a 2x2 matrix \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right|\) is given by:

\[
D = ad - bc
\]

For the given matrix, we have:
- \( a = -6 \)
- \( b = 9 \)
- \( c = x \)
- \( d = 5 \)

Thus, the determinant is:

\[
D = (-6)(5) - (9)(x) = -30 - 9x
\]

We are given the inequality \(D > 0\), so:

\[
-30 - 9x > 0
\]

Now, solve for \(x\):

\[
-30 > 9x
\]

\[
\frac{-30}{9} > x
\]

\[
x < \frac{-10}{3}
\]

Thus, the solution to the inequality is:

\[
x < \frac{-10}{3}
\]

Would you like a detailed explanation of each step?

Here are 5 related questions:
1. How do we solve the determinant of a 3x3 matrix inequality?
2. Can you explain the properties of determinants that affect inequalities?
3. What is the geometric interpretation of solving matrix inequalities?
4. How does the determinant change if we swap rows in a 2x2 matrix?
5. What are the conditions for the determinant of a matrix to be zero?

**Tip:** Always remember that switching two rows of a matrix changes the sign of its determinant.

Solve the inequality involving a 2x2 determinant: | -6  9 | > 0.

This problem involves solving a 2x2 matrix determinant inequality | -6  9 | > 0. The detailed steps include calculating the determinant and solving the inequality -30 - 9x > 0. Suitable for students in Grades 9-12.

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