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Answer :
To solve this problem, we need to find the ages of buildings B and D in terms of the age of building C, which is represented as [tex]\( x \)[/tex].
1. Determine the ages of the buildings:
- Building C is [tex]\( x \)[/tex] years old.
- Building B was built two years before building C, so Building B is [tex]\( x + 2 \)[/tex] years old.
- Building D was built two years before building B, so Building D is [tex]\( x + 4 \)[/tex] years old.
2. Express the condition as an inequality:
We are told that the product of the ages of Building B and Building D is at least 195. This leads to the inequality:
[tex]\[
(x + 2) \times (x + 4) \geq 195
\][/tex]
3. Expand the left-hand side:
- First, expand the expression [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[
(x + 2)(x + 4) = x^2 + 4x + 2x + 8
\][/tex]
- Combine like terms:
[tex]\[
x^2 + 6x + 8
\][/tex]
4. Form the inequality:
Now, substitute the expanded expression back into the inequality:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]
Therefore, the inequality that represents the given situation is:
[tex]\[
\boxed{x^2 + 6x + 8 \geq 195}
\][/tex]
This corresponds to option C.
1. Determine the ages of the buildings:
- Building C is [tex]\( x \)[/tex] years old.
- Building B was built two years before building C, so Building B is [tex]\( x + 2 \)[/tex] years old.
- Building D was built two years before building B, so Building D is [tex]\( x + 4 \)[/tex] years old.
2. Express the condition as an inequality:
We are told that the product of the ages of Building B and Building D is at least 195. This leads to the inequality:
[tex]\[
(x + 2) \times (x + 4) \geq 195
\][/tex]
3. Expand the left-hand side:
- First, expand the expression [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[
(x + 2)(x + 4) = x^2 + 4x + 2x + 8
\][/tex]
- Combine like terms:
[tex]\[
x^2 + 6x + 8
\][/tex]
4. Form the inequality:
Now, substitute the expanded expression back into the inequality:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]
Therefore, the inequality that represents the given situation is:
[tex]\[
\boxed{x^2 + 6x + 8 \geq 195}
\][/tex]
This corresponds to option C.
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