To determine the correct system of inequalities representing the given situation, let's break down the problem step by step:
1. Mail Capacity Constraint:
The distribution center processes up to 150,000 pieces of mail each day.
- Each load by a land carrier can carry 1800 pieces.
- Each load by an air carrier can carry 1500 pieces.
So, if [tex]\( x \)[/tex] represents the number of loads by land carriers and [tex]\( y \)[/tex] represents the number of loads by air carriers, the total number of pieces of mail carried can be represented by the expression [tex]\( 1800x + 1500y \)[/tex].
The center can process a maximum of 150,000 pieces of mail each day, so the inequality representing this constraint is:
[tex]\[
1800x + 1500y \leq 150,000
\][/tex]
2. Load Capacity Constraint:
The loading equipment can handle a maximum of 150 loads per day. This includes both land and air carriers combined.
Hence, the total number of loads [tex]\( x \)[/tex] (land) plus [tex]\( y \)[/tex] (air) must be less than or equal to 150:
[tex]\[
x + y \leq 150
\][/tex]
Putting these two inequalities together, the system that represents the given situation is:
[tex]\[
1800x + 1500y \leq 150,000
\][/tex]
[tex]\[
x + y \leq 150
\][/tex]
From the given options, the set of inequalities that matches this is:
[tex]\(1800 x + 1500 y \leq 150,000\)[/tex] and [tex]\(x + y \leq 150\)[/tex].
Thus, the correct choice is the fourth one:
[tex]\[
\boxed{1800 x + 1500 y \leq 150,000 \quad x + y \leq 150}
\][/tex]
