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A construction company is analyzing which of its older projects need renovation. Building B was built two years before building C. Building D was built two years before building B. The product of building B's age and building D's age is at least 195. If [tex]x[/tex] represents the age of building C, which inequality represents this situation?

A. [tex]x^2 + 6x + 8 \geq 195[/tex]

B. [tex]x^2 + 4x + 4 \geq 195[/tex]

C. [tex]x^2 + 8x + 16 \geq 195[/tex]

D. [tex]x^2 + 4 \geq 195[/tex]

Answer :

Let [tex]$x$[/tex] represent the age of building [tex]$C$[/tex]. According to the problem:

1. Building [tex]$B$[/tex] was built two years before building [tex]$C$[/tex], so its age is [tex]$x+2$[/tex].
2. Building [tex]$D$[/tex] was built two years before building [tex]$B$[/tex], so its age is [tex]$x+4$[/tex].

The product of the ages of buildings [tex]$B$[/tex] and [tex]$D$[/tex] is given by
[tex]$$
(x+2)(x+4).
$$[/tex]

Expanding this product:
[tex]$$
(x+2)(x+4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8.
$$[/tex]

The problem states that the product of the ages is at least [tex]$195$[/tex]:
[tex]$$
x^2 + 6x + 8 \geq 195.
$$[/tex]

This inequality matches option A.

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