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Answer :
- Building B's age is expressed as $x - 2$, where $x$ is building C's age.
- Building D's age is expressed as $x - 4$.
- The product of building B's and building D's ages is $(x - 2)(x - 4)$, which must be at least 195.
- Expanding the product gives the inequality $x^2 - 6x + 8 \geq 195$.
### Explanation
1. Understanding the Problem
Let's break down this word problem step by step to figure out the correct inequality. We're given information about the ages of three buildings (B, C, and D) and how they relate to each other. Our goal is to express the given condition—that the product of building B's age and building D's age is at least 195—as an inequality in terms of $x$, where $x$ represents the age of building C.
2. Expressing Building B's Age
We know that building B was built two years before building C. This means building B is younger than building C. If building C is $x$ years old, then building B is $x - 2$ years old.
3. Expressing Building D's Age
Next, we know that building D was built two years before building B. So, building D is younger than building B. Since building B is $x - 2$ years old, building D is $(x - 2) - 2 = x - 4$ years old.
4. Forming the Inequality
The problem states that the product of building B's age and building D's age is at least 195. This translates to the inequality $(x - 2)(x - 4) ">=" 195$.
5. Expanding the Inequality
Now, let's expand the left side of the inequality:
$(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8$.
So, the inequality becomes $x^2 - 6x + 8 ">=" 195$.
6. Identifying the Correct Option
Comparing our derived inequality $x^2 - 6x + 8 ">=" 195$ with the given options, we see that option A, $x^2 + 6x + 8 ">=" 195$, has a sign error. The correct inequality should have $-6x$, not $+6x$. Therefore, the correct answer is not A.
However, there seems to be a mistake in the options provided. The correct inequality we derived is $x^2 - 6x + 8 ">=" 195$, which is not present in the options. Let's re-examine the problem statement and our steps to ensure no errors were made.
7. Final Answer
Upon reviewing the steps, the correct inequality representing the situation is indeed $x^2 - 6x + 8 ">=" 195$. Since this option is not available, and there might be a typo in the question, we will proceed by solving the inequality to find a suitable answer. However, since the question asks to select the correct inequality, and none of the options match the derived inequality, it indicates a possible error in the provided choices.
8. Conclusion
The correct inequality that represents the situation is $x^2 - 6x + 8 ">=" 195$. However, since this is not one of the options, and assuming there was a typo and option A was meant to be $x^2 - 6x + 8 ">=" 195$, we would select that. But as it stands, none of the options are correct.
### Examples
Understanding inequalities is crucial in various real-world scenarios. For instance, construction companies use inequalities to determine the minimum amount of materials needed for a project while staying within budget constraints. Similarly, in finance, inequalities help in calculating the minimum return on investment required to meet financial goals. In logistics, they are used to optimize delivery routes, ensuring that travel time or distance stays below a certain threshold. These applications highlight how inequalities provide a framework for making informed decisions and achieving desired outcomes in practical situations.
- Building D's age is expressed as $x - 4$.
- The product of building B's and building D's ages is $(x - 2)(x - 4)$, which must be at least 195.
- Expanding the product gives the inequality $x^2 - 6x + 8 \geq 195$.
### Explanation
1. Understanding the Problem
Let's break down this word problem step by step to figure out the correct inequality. We're given information about the ages of three buildings (B, C, and D) and how they relate to each other. Our goal is to express the given condition—that the product of building B's age and building D's age is at least 195—as an inequality in terms of $x$, where $x$ represents the age of building C.
2. Expressing Building B's Age
We know that building B was built two years before building C. This means building B is younger than building C. If building C is $x$ years old, then building B is $x - 2$ years old.
3. Expressing Building D's Age
Next, we know that building D was built two years before building B. So, building D is younger than building B. Since building B is $x - 2$ years old, building D is $(x - 2) - 2 = x - 4$ years old.
4. Forming the Inequality
The problem states that the product of building B's age and building D's age is at least 195. This translates to the inequality $(x - 2)(x - 4) ">=" 195$.
5. Expanding the Inequality
Now, let's expand the left side of the inequality:
$(x - 2)(x - 4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8$.
So, the inequality becomes $x^2 - 6x + 8 ">=" 195$.
6. Identifying the Correct Option
Comparing our derived inequality $x^2 - 6x + 8 ">=" 195$ with the given options, we see that option A, $x^2 + 6x + 8 ">=" 195$, has a sign error. The correct inequality should have $-6x$, not $+6x$. Therefore, the correct answer is not A.
However, there seems to be a mistake in the options provided. The correct inequality we derived is $x^2 - 6x + 8 ">=" 195$, which is not present in the options. Let's re-examine the problem statement and our steps to ensure no errors were made.
7. Final Answer
Upon reviewing the steps, the correct inequality representing the situation is indeed $x^2 - 6x + 8 ">=" 195$. Since this option is not available, and there might be a typo in the question, we will proceed by solving the inequality to find a suitable answer. However, since the question asks to select the correct inequality, and none of the options match the derived inequality, it indicates a possible error in the provided choices.
8. Conclusion
The correct inequality that represents the situation is $x^2 - 6x + 8 ">=" 195$. However, since this is not one of the options, and assuming there was a typo and option A was meant to be $x^2 - 6x + 8 ">=" 195$, we would select that. But as it stands, none of the options are correct.
### Examples
Understanding inequalities is crucial in various real-world scenarios. For instance, construction companies use inequalities to determine the minimum amount of materials needed for a project while staying within budget constraints. Similarly, in finance, inequalities help in calculating the minimum return on investment required to meet financial goals. In logistics, they are used to optimize delivery routes, ensuring that travel time or distance stays below a certain threshold. These applications highlight how inequalities provide a framework for making informed decisions and achieving desired outcomes in practical situations.
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