The average annual energy cost for a certain home is $4,334. The homeowner plans to spend $25,000 to install a...

1. TRANSLATE the problem information

• Given information:
  • Current annual energy cost: \(\$4,334\)
  • Installation cost: \(\$25,000\)
  • New annual energy cost: \(\$2,712\)
  • Want inequality for t years where total savings exceed installation cost

2. INFER what "savings" means

• Annual savings = Old cost - New cost
• Annual savings = \(\$4,334 - \$2,712 = \$1,622\)
• This matches the expression \(\mathrm{(4,334 - 2,712)}\) in the answer choices

3. INFER how total savings accumulate over time

• After t years, total savings = Annual savings × t
• Total savings = \(\mathrm{(4,334 - 2,712)t}\)

4. TRANSLATE "exceed the installation cost"

• "Exceed" means "greater than"
• We want: Total savings > Installation cost
• So: \(\mathrm{(4,334 - 2,712)t \gt 25,000}\)
• This can be rewritten as: \(\mathrm{25,000 \lt (4,334 - 2,712)t}\)

Answer: B. \(\mathrm{25,000 \lt (4,334 - 2,712)t}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "savings will exceed installation cost" and set up the inequality backwards.

They think: "Installation cost is greater than savings" and write \(\mathrm{25,000 \gt (4,334 - 2,712)t}\)

This may lead them to select Choice A (\(\mathrm{25,000 \gt (4,334 - 2,712)t}\))

Second Most Common Error:

Poor INFER reasoning: Students don't properly understand how to calculate savings or set up the time component.

They might subtract the annual cost from the installation cost instead of recognizing that savings accumulate annually. This leads to misunderstanding the structure entirely.

This may lead them to select Choice C (\(\mathrm{25,000 - 4,334 \gt 2,712t}\))

The Bottom Line:

The key challenge is correctly translating "exceed" into the proper inequality direction and understanding that savings accumulate over multiple years through multiplication, not just simple subtraction.