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

Sagot :

GinnyAnsweridn
Let's solve this step-by-step:

1. Let's denote Caitlyn's age as [tex]\( x \)[/tex].

2. Carl is two years older than Caitlyn, so Carl's age can be represented as [tex]\( x + 2 \)[/tex].

3. Daryl is five years older than Carl, so Daryl's age can be represented as [tex]\( (x + 2) + 5 \)[/tex], which simplifies to [tex]\( x + 7 \)[/tex].

4. Now, the given problem states that the product of Carl's and Daryl's ages is at least 160. Thus the inequality can be written as:
[tex]\[ (x + 2)(x + 7) \geq 160 \][/tex]

Let's expand the product to better understand the inequality:
[tex]\[ (x + 2)(x + 7) = x^2 + 7x + 2x + 14 = x^2 + 9x + 14 \][/tex]

So the inequality representing the situation is:
[tex]\[ x^2 + 9x + 14 \geq 160 \][/tex]

The correct answer is:
[tex]\[ x^2 + 9x + 14 \geq 160 \][/tex]
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