To find the value of [tex]\( h(-6) \)[/tex] for the given piecewise function [tex]\( h(x) \)[/tex], let's follow the steps methodically.
The piecewise function is defined as follows:
[tex]\[
h(x) =
\begin{cases}
-4x + 2 & \text{if } x \leq -5 \\
x + 7 & \text{if } -5 < x < 5 \\
3x - 5 & \text{if } x \geq 5
\end{cases}
\][/tex]
Step 1: Identify the correct interval for [tex]\( x = -6 \)[/tex].
We need to determine which case of the piecewise function applies to [tex]\( x = -6 \)[/tex]:
[tex]\[
x \leq -5
\][/tex]
Since [tex]\( -6 \)[/tex] is less than or equal to [tex]\( -5 \)[/tex], we use the first part of the piecewise function:
[tex]\[
h(x) = -4x + 2
\][/tex]
Step 2: Substitute [tex]\( x = -6 \)[/tex] into the appropriate piece of the function.
[tex]\[
h(-6) = -4(-6) + 2
\][/tex]
Step 3: Perform the arithmetic operations:
[tex]\[
-4(-6) = 24
\][/tex]
[tex]\[
24 + 2 = 26
\][/tex]
So, the value of [tex]\( h(-6) \)[/tex] is:
[tex]\[
h(-6) = 26
\][/tex]
Thus, the correct answer is [tex]\(\boxed{26}\)[/tex].
