High School

Select the correct answer.

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 :

To solve the problem, let's follow these steps:

1. Understand the relationship between the buildings and their ages:
- Building C's age is represented by [tex]\( x \)[/tex].
- Building B was built two years before Building C, so the age of Building B is [tex]\( x - 2 \)[/tex].
- Building D was built two years before Building B, therefore the age of Building D is [tex]\( x - 4 \)[/tex].

2. Translate the information into an inequality:
- The problem states that the product of the ages of Building B and Building D is at least 195. This can be expressed as:
[tex]\[
(x - 2) \times (x - 4) \geq 195
\][/tex]

3. Simplify and expand the inequality:
- Distribute the expression:
[tex]\[
(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8
\][/tex]

4. Set up the inequality:
- Now we have:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]

The inequality representing the situation is:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]

The correct answer choice is not listed in the given options, but based on our understanding, this is the resulting expression from translating the given problem into a mathematical expression. Thus, you should use the process to understand and verify the relationships to arrive at the correct inequality.