Select the correct answer.
A construction company is analyzing which of its older projects need renovation. Building $B$ was built two years before building $C$. Building $D$ was built two years before building $B$. The product of building $B$'s age and building $D$'s age is at least 195. If $x$ represents the age of building $C$, which inequality represents this situation?
A. $x^2+8 x+16 \geq 195$
B. $x^2+4 x+4 \geq 195$
C. $x^2+4 \geq 195$
D. $x^2+6 x+8 \geq 195$
Answer (1)
Express building B's age as x − 2 and building D's age as x − 4 .
Formulate the inequality ( x − 2 ) ( x − 4 ) g e 195 .
Expand the expression to get x 2 − 6 x + 8 g e 195 .
The correct inequality is x 2 − 6 x + 8 ≥ 195 .
Explanation
Analyze the problem Let's analyze the given information to form an inequality. We know the age of building C is x . Building B was built two years before building C, so building B's age is x − 2 . Building D was built two years before building B, so building D's age is ( x − 2 ) − 2 = x − 4 . The product of building B's age and building D's age is at least 195, which means ( x − 2 ) ( x − 4 ) ≥ 195 .
Expand the expression Now, let's expand the expression ( x − 2 ) ( x − 4 ) .
( x − 2 ) ( x − 4 ) = x 2 − 4 x − 2 x + 8 = x 2 − 6 x + 8 .
Form the inequality So, the inequality is x 2 − 6 x + 8 ≥ 195 . Comparing this with the given options, we see that option D matches our derived inequality.
State the answer Therefore, the correct inequality that represents the situation is x 2 − 6 x + 8 ≥ 195 .
Examples
Consider a scenario where a construction company needs to determine which buildings require renovation based on their age. If the age of Building C is represented by 'x', and we know the relationships between the ages of Buildings B and D relative to Building C, we can formulate an inequality to model the condition that the product of the ages of Buildings B and D must be at least 195 years. This inequality helps the company identify buildings that meet this criterion, aiding in their renovation planning and resource allocation. For example, if x = 18, then building B is 16 years old and building D is 14 years old. The product of their ages is 224, which is greater than 195.
