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Answer :
To solve the problem, we need to establish the ages of the buildings based on the given relationships and then form an inequality representing the condition about the buildings' ages.
1. Identify the ages of the buildings:
- Let [tex]\( x \)[/tex] represent 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, thus making Building D's age [tex]\( x + 4 \)[/tex] years.
2. Formulate the condition based on the problem:
- We are given that the product of Building B's age and Building D's age must be at least 195.
- This can be written as: [tex]\((x + 2) \times (x + 4) \geq 195\)[/tex].
3. Expand and simplify the inequality:
- Multiply the terms: [tex]\((x + 2)(x + 4)\)[/tex].
- Use the distributive property to expand: [tex]\(x \cdot (x + 4) + 2 \cdot (x + 4)\)[/tex].
- This equals: [tex]\(x^2 + 4x + 2x + 8\)[/tex].
- Combine like terms: [tex]\(x^2 + 6x + 8\)[/tex].
4. Write the final inequality:
- The expanded expression [tex]\((x^2 + 6x + 8)\)[/tex] must be at least 195.
- Therefore, the inequality representing the situation is: [tex]\[x^2 + 6x + 8 \geq 195.\][/tex]
Based on these steps, the correct answer is:
D. [tex]\(x^2 + 6x + 8 \geq 195\)[/tex]
1. Identify the ages of the buildings:
- Let [tex]\( x \)[/tex] represent 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, thus making Building D's age [tex]\( x + 4 \)[/tex] years.
2. Formulate the condition based on the problem:
- We are given that the product of Building B's age and Building D's age must be at least 195.
- This can be written as: [tex]\((x + 2) \times (x + 4) \geq 195\)[/tex].
3. Expand and simplify the inequality:
- Multiply the terms: [tex]\((x + 2)(x + 4)\)[/tex].
- Use the distributive property to expand: [tex]\(x \cdot (x + 4) + 2 \cdot (x + 4)\)[/tex].
- This equals: [tex]\(x^2 + 4x + 2x + 8\)[/tex].
- Combine like terms: [tex]\(x^2 + 6x + 8\)[/tex].
4. Write the final inequality:
- The expanded expression [tex]\((x^2 + 6x + 8)\)[/tex] must be at least 195.
- Therefore, the inequality representing the situation is: [tex]\[x^2 + 6x + 8 \geq 195.\][/tex]
Based on these steps, the correct answer is:
D. [tex]\(x^2 + 6x + 8 \geq 195\)[/tex]
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