Sure, let's start by translating the ages into algebraic expressions based on the information provided:
1. Carl's age: Let [tex]\( x \)[/tex] be Caitlyn's age. Since Carl is 2 years older than Caitlyn, Carl's age is [tex]\( x + 2 \)[/tex].
2. Daryl's age: Since Daryl is 5 years older than Carl, Daryl’s age is [tex]\( (x + 2) + 5 = x + 7 \)[/tex].
The next step involves setting up the inequality based on the product of Carl and Daryl's ages, which needs to be at least 160:
[tex]\[
(x + 2)(x + 7) \geq 160
\][/tex]
Now, let's expand this product:
1. First, distribute [tex]\( x + 2 \)[/tex] and [tex]\( x + 7 \)[/tex]:
[tex]\[
(x + 2)(x + 7) = x^2 + 7x + 2x + 14
\][/tex]
2. Combine like terms:
[tex]\[
x^2 + 7x + 2x + 14 = x^2 + 9x + 14
\][/tex]
So, the inequality simplifies to:
[tex]\[
x^2 + 9x + 14 \geq 160
\][/tex]
The correct inequality that represents this situation is:
[tex]\[
x^2 + 9x + 14 \geq 160
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x^2 + 9x + 14 \geq 160}
\][/tex]
