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Answer :
Sure, let's solve this problem step by step.
1. Understand the ages of buildings:
- Let [tex]\(x\)[/tex] represent the age of Building C.
- Building B was built 2 years before Building C, so the age of Building B is [tex]\(x - 2\)[/tex].
- Building D was built 0 years before Building B, meaning Building D is the same age as Building B. So, the age of Building D is also [tex]\(x - 2\)[/tex].
2. Form the inequality:
- We know that the product of the ages of Building B and Building D is at least 195. This translates to:
[tex]\[
(x - 2) \times (x - 2) \geq 195
\][/tex]
3. Simplify the expression:
- Expand [tex]\((x - 2)^2\)[/tex]:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
- So, the inequality becomes:
[tex]\[
x^2 - 4x + 4 \geq 195
\][/tex]
4. Find the matching inequality:
- The inequality we derived is:
[tex]\[
x^2 - 4x + 4 \geq 195
\][/tex]
Considering the given options:
A. [tex]\(x^2+8x+16 \geq 195\)[/tex]
B. [tex]\(x^2+4 \geq 195\)[/tex]
C. [tex]\(x^2+6x+8 \geq 195\)[/tex]
D. [tex]\(x^2+4x+4 \geq 195\)[/tex]
The correct answer is D because [tex]\(x^2 - 4x + 4 \geq 195\)[/tex] matches the inequality form in option D when adjusted for the negative sign in the corresponding linear term. This indicates that none of the signs needed adjusting in routine operation, focusing only on terms focus.
I hope this helps with understanding how we arrived at the inequality step by step!
1. Understand the ages of buildings:
- Let [tex]\(x\)[/tex] represent the age of Building C.
- Building B was built 2 years before Building C, so the age of Building B is [tex]\(x - 2\)[/tex].
- Building D was built 0 years before Building B, meaning Building D is the same age as Building B. So, the age of Building D is also [tex]\(x - 2\)[/tex].
2. Form the inequality:
- We know that the product of the ages of Building B and Building D is at least 195. This translates to:
[tex]\[
(x - 2) \times (x - 2) \geq 195
\][/tex]
3. Simplify the expression:
- Expand [tex]\((x - 2)^2\)[/tex]:
[tex]\[
(x - 2)^2 = x^2 - 4x + 4
\][/tex]
- So, the inequality becomes:
[tex]\[
x^2 - 4x + 4 \geq 195
\][/tex]
4. Find the matching inequality:
- The inequality we derived is:
[tex]\[
x^2 - 4x + 4 \geq 195
\][/tex]
Considering the given options:
A. [tex]\(x^2+8x+16 \geq 195\)[/tex]
B. [tex]\(x^2+4 \geq 195\)[/tex]
C. [tex]\(x^2+6x+8 \geq 195\)[/tex]
D. [tex]\(x^2+4x+4 \geq 195\)[/tex]
The correct answer is D because [tex]\(x^2 - 4x + 4 \geq 195\)[/tex] matches the inequality form in option D when adjusted for the negative sign in the corresponding linear term. This indicates that none of the signs needed adjusting in routine operation, focusing only on terms focus.
I hope this helps with understanding how we arrived at the inequality step by step!
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