Select the correct answer.

Carl, Caitlyn, and Daryl are comparing their ages. Carl is two years older than Caitlyn. Daryl is five years older than Carl. The product of Carl's and Daryl's ages is at least 160. If [tex]x[/tex] represents Caitlyn's age, which inequality represents this situation?

A. [tex]x^2 + 9x + 14 \geq 160[/tex]
B. [tex]x^2 + 2x + 5 \geq 160[/tex]
C. [tex]x^2 + 14x + 28 \geq 160[/tex]
D. [tex]x^2 + 4 \geq 160[/tex]



Answer :

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]