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Answer :
Let's solve the problem step-by-step:
1. Identify the ages of the buildings in terms of [tex]\( x \)[/tex]:
- 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].
- Building D was built two years before building B, so the age of building D is [tex]\( (x + 2) + 2 = x + 4 \)[/tex].
2. Formulate the inequality:
- The problem states that the product of building B's age and building D's age is at least 195.
- Therefore, the inequality can be expressed as:
[tex]\[
(x + 2)(x + 4) \geq 195
\][/tex]
3. Expand the expression:
- Expand [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[
(x + 2)(x + 4) = x(x + 4) + 2(x + 4)
\][/tex]
[tex]\[
= x^2 + 4x + 2x + 8
\][/tex]
[tex]\[
= x^2 + 6x + 8
\][/tex]
4. Set up the inequality:
- The expanded expression forms the inequality:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]
5. Select the correct option:
- Among the given options, option C ([tex]\(x^2 + 6x + 8 \geq 195\)[/tex]) correctly represents the formulated inequality.
Therefore, the correct answer is:
[tex]\[ \text{C. } x^2 + 6x + 8 \geq 195 \][/tex]
1. Identify the ages of the buildings in terms of [tex]\( x \)[/tex]:
- 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].
- Building D was built two years before building B, so the age of building D is [tex]\( (x + 2) + 2 = x + 4 \)[/tex].
2. Formulate the inequality:
- The problem states that the product of building B's age and building D's age is at least 195.
- Therefore, the inequality can be expressed as:
[tex]\[
(x + 2)(x + 4) \geq 195
\][/tex]
3. Expand the expression:
- Expand [tex]\((x + 2)(x + 4)\)[/tex]:
[tex]\[
(x + 2)(x + 4) = x(x + 4) + 2(x + 4)
\][/tex]
[tex]\[
= x^2 + 4x + 2x + 8
\][/tex]
[tex]\[
= x^2 + 6x + 8
\][/tex]
4. Set up the inequality:
- The expanded expression forms the inequality:
[tex]\[
x^2 + 6x + 8 \geq 195
\][/tex]
5. Select the correct option:
- Among the given options, option C ([tex]\(x^2 + 6x + 8 \geq 195\)[/tex]) correctly represents the formulated inequality.
Therefore, the correct answer is:
[tex]\[ \text{C. } x^2 + 6x + 8 \geq 195 \][/tex]
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