To solve this problem, let's denote Caitlyn's age as [tex]\( x \)[/tex].
1. Determine Carl's Age:
Carl is two years older than Caitlyn. Thus, Carl's age is:
[tex]\[
x + 2
\][/tex]
2. Determine Daryl's Age:
Daryl is five years older than Carl. Thus, Daryl's age is:
[tex]\[
(x + 2) + 5 = x + 7
\][/tex]
3. Set Up the Inequality:
The product of Carl's and Daryl's ages is at least 160. Therefore, we have:
[tex]\[
(x + 2)(x + 7) \geq 160
\][/tex]
4. Expand the Inequality:
Multiply the expressions:
[tex]\[
(x + 2)(x + 7) = x(x + 7) + 2(x + 7)
\][/tex]
Simplify each term:
[tex]\[
= x^2 + 7x + 2x + 14
\][/tex]
Combine like terms:
[tex]\[
= x^2 + 9x + 14
\][/tex]
5. Formulate the Final Inequality:
We need this product to be at least 160:
[tex]\[
x^2 + 9x + 14 \geq 160
\][/tex]
Therefore, the correct inequality that represents this situation is:
[tex]\[
x^2 + 9x + 14 \geq 160
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{x^2 + 9x + 14 \geq 160}
\][/tex]
