Solve the compound inequality:

[tex]\[ 6-x\ \textgreater \ 15 \text{ or } 2x-9 \geq 3 \][/tex]

A. [tex]\( x \ \textgreater \ 9 \)[/tex] or [tex]\( x \leq 6 \)[/tex]
B. [tex]\( x \ \textless \ 9 \)[/tex] or [tex]\( x \geq -6 \)[/tex]
C. [tex]\( x \ \textgreater \ -9 \)[/tex] or [tex]\( x \leq -6 \)[/tex]
D. [tex]\( x \ \textless \ -9 \)[/tex] or [tex]\( x \geq 6 \)[/tex]

Let's solve the given compound inequality step-by-step:

We have two inequalities to solve:

1. [tex]\(6 - x > 15\)[/tex]
2. [tex]\(2x - 9 \geq 3\)[/tex]

Let's solve each inequality separately:

For the first inequality, [tex]\(6 - x > 15\)[/tex]:

- Subtract 6 from both sides:
[tex]\[
-x > 9
\][/tex]

- Now, multiply both sides by -1. Remember, multiplying or dividing by a negative number reverses the inequality sign:
[tex]\[
x < -9
\][/tex]

For the second inequality, [tex]\(2x - 9 \geq 3\)[/tex]:

- Add 9 to both sides:
[tex]\[
2x \geq 12
\][/tex]

- Divide both sides by 2:
[tex]\[
x \geq 6
\][/tex]

Combining the solutions:

The solutions from both inequalities are [tex]\(x < -9\)[/tex] and [tex]\(x \geq 6\)[/tex]. Since this is a compound inequality using the word "or," it means that [tex]\(x\)[/tex] can satisfy either one of the two conditions.

So, the solution to the compound inequality is:
[tex]\[ x < -9 \][/tex] or [tex]\[ x \geq 6 \][/tex]

Based on the provided choices, the correct answer is D: [tex]\(x < -9\)[/tex] or [tex]\(x \geq 6\)[/tex].