Answer :
- Express building B's age as $x - 2$ and building D's age as $x - 4$.
- Form the inequality $(x - 2)(x - 4) \geq 195$.
- Expand and simplify the inequality to $x^2 - 6x + 8 \geq 195$.
- The correct inequality representing the situation is $\boxed{x^2 - 6x + 8 \geq 195}$.
### Explanation
1. Problem Analysis
Let's analyze the given information to form an inequality.
We are given that:
1. Building B was built two years before building C.
2. Building D was built two years before building B.
3. The product of building B's age and building D's age is at least 195.
4. x represents the age of building C.
We need to find the inequality that represents this situation in terms of x (age of building C).
2. Expressing Ages in Terms of x
Let's express the ages of buildings B and D in terms of x.
Since building B was built two years before building C, the age of building B is $x - 2$.
Since building D was built two years before building B, the age of building D is $(x - 2) - 2 = x - 4$.
3. Forming the Inequality
Now, let's write the inequality representing the product of building B's age and building D's age being at least 195:
$(x - 2)(x - 4) \geq 195$
4. Expanding and Simplifying
Expand the left side of the inequality:
$x^2 - 4x - 2x + 8 \geq 195$
Simplify the inequality:
$x^2 - 6x + 8 \geq 195$
5. Selecting the Correct Answer
Comparing the derived inequality with the given options, we find that option D matches our result.
Therefore, the correct answer is D. $x^2 - 6x + 8 \geq 195$
### Examples
Understanding inequalities like this can help in various real-world scenarios. For instance, if a company wants to ensure its profits are above a certain threshold, they can model their revenue and costs as expressions involving variables like sales volume or production costs. By setting up an inequality, they can determine the minimum sales volume needed to achieve the desired profit level. This type of analysis is crucial for making informed business decisions and setting realistic goals.
- Form the inequality $(x - 2)(x - 4) \geq 195$.
- Expand and simplify the inequality to $x^2 - 6x + 8 \geq 195$.
- The correct inequality representing the situation is $\boxed{x^2 - 6x + 8 \geq 195}$.
### Explanation
1. Problem Analysis
Let's analyze the given information to form an inequality.
We are given that:
1. Building B was built two years before building C.
2. Building D was built two years before building B.
3. The product of building B's age and building D's age is at least 195.
4. x represents the age of building C.
We need to find the inequality that represents this situation in terms of x (age of building C).
2. Expressing Ages in Terms of x
Let's express the ages of buildings B and D in terms of x.
Since building B was built two years before building C, the age of building B is $x - 2$.
Since building D was built two years before building B, the age of building D is $(x - 2) - 2 = x - 4$.
3. Forming the Inequality
Now, let's write the inequality representing the product of building B's age and building D's age being at least 195:
$(x - 2)(x - 4) \geq 195$
4. Expanding and Simplifying
Expand the left side of the inequality:
$x^2 - 4x - 2x + 8 \geq 195$
Simplify the inequality:
$x^2 - 6x + 8 \geq 195$
5. Selecting the Correct Answer
Comparing the derived inequality with the given options, we find that option D matches our result.
Therefore, the correct answer is D. $x^2 - 6x + 8 \geq 195$
### Examples
Understanding inequalities like this can help in various real-world scenarios. For instance, if a company wants to ensure its profits are above a certain threshold, they can model their revenue and costs as expressions involving variables like sales volume or production costs. By setting up an inequality, they can determine the minimum sales volume needed to achieve the desired profit level. This type of analysis is crucial for making informed business decisions and setting realistic goals.
