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Answer :
To solve this problem, let's follow the given information step-by-step:
1. Identify the ages of the buildings:
- Let [tex]\( x \)[/tex] be the age of building C.
- Building B was built two years before building C, so the age of building B is [tex]\( x + 2 \)[/tex] years.
- Building D was built two years before building B, so the age of building D is [tex]\( x + 4 \)[/tex] years.
2. Formulate the product of their ages:
- The product of the ages of building B and building D is given as [tex]\((x + 2) \times (x + 4)\)[/tex].
3. Set up the inequality:
- The product is at least 195, so we write this as:
[tex]\[(x + 2)(x + 4) \geq 195\][/tex]
4. Expand the left side of the inequality:
- Distribute to simplify the expression:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
Therefore, the inequality becomes:
[tex]\[ x^2 + 6x + 8 \geq 195 \][/tex]
5. Identify the correct option:
- Finally, we compare this inequality to the given answer choices:
- A. [tex]\( x^2 + 4x + 4 \geq 195 \)[/tex]
- B. [tex]\( x^2 + 4 \geq 195 \)[/tex]
- C. [tex]\( x^2 + 8x + 16 \geq 195 \)[/tex]
- D. [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex]
The correct inequality that represents the situation is:
[tex]\[ \boxed{x^2 + 6x + 8 \geq 195} \][/tex]
1. Identify the ages of the buildings:
- Let [tex]\( x \)[/tex] be the age of building C.
- Building B was built two years before building C, so the age of building B is [tex]\( x + 2 \)[/tex] years.
- Building D was built two years before building B, so the age of building D is [tex]\( x + 4 \)[/tex] years.
2. Formulate the product of their ages:
- The product of the ages of building B and building D is given as [tex]\((x + 2) \times (x + 4)\)[/tex].
3. Set up the inequality:
- The product is at least 195, so we write this as:
[tex]\[(x + 2)(x + 4) \geq 195\][/tex]
4. Expand the left side of the inequality:
- Distribute to simplify the expression:
[tex]\[ (x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8 \][/tex]
Therefore, the inequality becomes:
[tex]\[ x^2 + 6x + 8 \geq 195 \][/tex]
5. Identify the correct option:
- Finally, we compare this inequality to the given answer choices:
- A. [tex]\( x^2 + 4x + 4 \geq 195 \)[/tex]
- B. [tex]\( x^2 + 4 \geq 195 \)[/tex]
- C. [tex]\( x^2 + 8x + 16 \geq 195 \)[/tex]
- D. [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex]
The correct inequality that represents the situation is:
[tex]\[ \boxed{x^2 + 6x + 8 \geq 195} \][/tex]
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