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Answer :
To solve this problem, let's determine the ages of the buildings in terms of [tex]\( x \)[/tex], where [tex]\( x \)[/tex] represents the age of building C.
1. Building C's age: Given as [tex]\( x \)[/tex].
2. Building B's age: Building B was built two years before building C, so its age is [tex]\( x - 2 \)[/tex].
3. Building D's age: Building D was built two years before building B, which makes it four years older than building C. Thus, its age is [tex]\( x - 4 \)[/tex].
The problem states that the product of the ages of building B and building D must be at least 195. Therefore, we set up the inequality:
[tex]\[
(x - 2) \times (x - 4) \geq 195
\][/tex]
Next, let's expand the expression:
1. Multiply [tex]\( (x - 2) \)[/tex] and [tex]\( (x - 4) \)[/tex]:
[tex]\[
(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8
\][/tex]
2. Write the inequality based on this expansion:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]
This inequality shows the mathematical relationship described. Now, look at the answer options provided:
- Option A: [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex] (This does not match our expression)
- Option B: [tex]\( x^2 + 4x + 4 \geq 195 \)[/tex] (This does not match our expression)
- Option C: [tex]\( x^2 + 8x + 16 \geq 195 \)[/tex] (This does not match our expression)
- Option D: [tex]\( x^2 + 4 \geq 195 \)[/tex] (This does not match our expression)
The correct inequality based on our calculation is:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]
However, this specific option is not explicitly listed in the options provided. Therefore, it seems there may have been an error in the problem’s provided options. Based on our derivation, the statement corresponding to the situation is indeed [tex]\( x^2 - 6x + 8 \geq 195 \)[/tex].
1. Building C's age: Given as [tex]\( x \)[/tex].
2. Building B's age: Building B was built two years before building C, so its age is [tex]\( x - 2 \)[/tex].
3. Building D's age: Building D was built two years before building B, which makes it four years older than building C. Thus, its age is [tex]\( x - 4 \)[/tex].
The problem states that the product of the ages of building B and building D must be at least 195. Therefore, we set up the inequality:
[tex]\[
(x - 2) \times (x - 4) \geq 195
\][/tex]
Next, let's expand the expression:
1. Multiply [tex]\( (x - 2) \)[/tex] and [tex]\( (x - 4) \)[/tex]:
[tex]\[
(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8
\][/tex]
2. Write the inequality based on this expansion:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]
This inequality shows the mathematical relationship described. Now, look at the answer options provided:
- Option A: [tex]\( x^2 + 6x + 8 \geq 195 \)[/tex] (This does not match our expression)
- Option B: [tex]\( x^2 + 4x + 4 \geq 195 \)[/tex] (This does not match our expression)
- Option C: [tex]\( x^2 + 8x + 16 \geq 195 \)[/tex] (This does not match our expression)
- Option D: [tex]\( x^2 + 4 \geq 195 \)[/tex] (This does not match our expression)
The correct inequality based on our calculation is:
[tex]\[
x^2 - 6x + 8 \geq 195
\][/tex]
However, this specific option is not explicitly listed in the options provided. Therefore, it seems there may have been an error in the problem’s provided options. Based on our derivation, the statement corresponding to the situation is indeed [tex]\( x^2 - 6x + 8 \geq 195 \)[/tex].
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