To solve the given problem, we need to model the situation using linear inequalities. Let's break down the problem into smaller parts:
1. Maximum Working Hours:
- You can work a maximum of 40 hours a week.
- Let [tex]\( x \)[/tex] represent the number of hours worked at the office job.
- Let [tex]\( y \)[/tex] represent the number of hours worked at the babysitting job.
- The total hours worked in a week from both jobs combined must be less than or equal to 40 hours.
- This can be represented by the inequality:
[tex]\[
x + y \leq 40
\][/tex]
2. Minimum Earnings:
- You need to earn at least \[tex]$400 to cover your expenses.
- The office job pays \$[/tex]12 per hour.
- The babysitting job pays \[tex]$10 per hour.
- The total earnings from both jobs combined must be at least \$[/tex]400.
- The earnings from the office job can be calculated as [tex]\( 12x \)[/tex].
- The earnings from the babysitting job can be calculated as [tex]\( 10y \)[/tex].
- Therefore, the total earnings should be:
[tex]\[
12x + 10y \geq 400
\][/tex]
Now we have the two inequalities that model the situation:
1. [tex]\( x + y \leq 40 \)[/tex]
2. [tex]\( 12x + 10y \geq 400 \)[/tex]
Comparing these with the given options:
(A) [tex]\( x + y \leq 40, 12x + 10y \geq 400 \)[/tex] - This matches our findings.
(B) [tex]\( x + y \geq 40, 12x + 10y \leq 400 \)[/tex] - This does not match our findings because the first inequality should be [tex]\(\leq 40\)[/tex] and the second should be [tex]\(\geq 400\)[/tex].
(C) [tex]\( x - y \leq 40, 12x - 10y \geq 400 \)[/tex] - This does not match our findings because the structure of these inequalities is incorrect.
(D) [tex]\( x - y \geq 40, 12x - 10y \leq 400 \)[/tex] - This also does not match our findings due to incorrect inequality expressions.
Thus, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
