To determine whether the point [tex]\((-4, -6)\)[/tex] satisfies the system of inequalities
[tex]\[
\left\{\begin{array}{l}
y \leq 3x + 2 \\
y > x - 1
\end{array}\right.
\][/tex]
we need to check if this point satisfies both inequalities.
1. First inequality: [tex]\( y \leq 3x + 2 \)[/tex]
Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
-6 \leq 3(-4) + 2
\][/tex]
Simplify the right-hand side:
[tex]\[
-6 \leq -12 + 2
\][/tex]
[tex]\[
-6 \leq -10
\][/tex]
This is false since [tex]\(-6\)[/tex] is not less than or equal to [tex]\(-10\)[/tex].
2. Second inequality: [tex]\( y > x - 1 \)[/tex]
Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -6\)[/tex]:
[tex]\[
-6 > -4 - 1
\][/tex]
Simplify the right-hand side:
[tex]\[
-6 > -5
\][/tex]
This is false since [tex]\(-6\)[/tex] is not greater than [tex]\(-5\)[/tex].
Since the point [tex]\((-4, -6)\)[/tex] does not satisfy either of the inequalities, it does not satisfy the system of inequalities.
Therefore, the answer is [tex]\(\boxed{False}\)[/tex].
