To solve the given compound inequalities, we'll address each inequality separately and then combine the results. Here are the steps:
### First Inequality: [tex]\( 12x + 7 < -11 \)[/tex]
1. Subtract 7 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
12x + 7 - 7 < -11 - 7
\][/tex]
[tex]\[
12x < -18
\][/tex]
2. Divide both sides by 12 to solve for [tex]\( x \)[/tex]:
[tex]\[
x < \frac{-18}{12}
\][/tex]
[tex]\[
x < -\frac{3}{2}
\][/tex]
### Second Inequality: [tex]\( 5x - 8 \geq 40 \)[/tex]
1. Add 8 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
5x - 8 + 8 \geq 40 + 8
\][/tex]
[tex]\[
5x \geq 48
\][/tex]
2. Divide both sides by 5 to solve for [tex]\( x \)[/tex]:
[tex]\[
x \geq \frac{48}{5}
\][/tex]
[tex]\[
x \geq 9.6
\][/tex]
### Combining the Solutions
The solutions to the compound inequalities are [tex]\( x < -\frac{3}{2} \)[/tex] or [tex]\( x \geq 9.6 \)[/tex]. Since the solutions do not overlap and cover separate regions on the number line, we use the logical "or" to combine them.
Therefore, the correct solution is:
(A) [tex]\( x < -\frac{3}{2} \)[/tex] or [tex]\( x \geq \frac{48}{5} \)[/tex]
