Question 132·Hard·Linear Inequalities in One or Two Variables
A manufacturing plant aims to reduce its monthly energy consumption by at least 15% compared with its average monthly usage of 160,000 kilowatt-hours (kWh). During the first 14 days of April, the plant has already used 72,000 kWh. If represents the maximum number of kilowatt-hours the plant can still use during the remainder of April and meet its reduction goal, which of the following inequalities must be true?
For inequality word problems, first translate all verbal conditions into clear math pieces: identify the original amount, the required percentage change, and the total expression involving the variable. Convert phrases like "at least 15% reduction" into a maximum allowed usage (here, times the original). Then express the total in terms of the variable (include what has already happened plus what remains) and choose the inequality sign based on whether the situation requires staying below, above, or at a certain threshold. Finally, match your inequality to the answer choices exactly, paying close attention to which quantity is on each side and the direction of the inequality symbol.
Hints
Focus on what “at least 15% reduction” means
If the plant normally uses 160,000 kWh, and it wants to reduce usage by at least 15%, think: what percentage of 160,000 is it allowed to use at most?
Write the total April usage in terms of k
The plant has already used 72,000 kWh and will use more. What expression represents the total energy it will use in April?
Choose the correct inequality sign
To meet the goal, should the total usage be less than, greater than, or equal to the maximum allowed usage? That tells you whether to use , , , or between your two expressions.
Desmos Guide
Compute the percentage amounts
In Desmos, type 0.85*160000 to see the maximum allowed monthly usage and 0.15*160000 to see the required reduction amount. Note which number represents how much they can use and which represents how much they must save.
Relate total usage to the allowed maximum
Enter k+72000 as an expression to represent total monthly usage. Think about whether this expression should be less than or equal to, greater than, or greater than or equal to the value you got from 0.85*160000 in order to satisfy the goal; then see which answer choice uses that same comparison.
Step-by-step Explanation
Translate the percentage reduction into a usage limit
"Reduce its monthly energy consumption by at least 15%" means the plant must save at least 15% of 160,000 kWh.
So the plant can use at most the remaining 85% of its usual amount:
(You could calculate this number, but for matching the answer choices, it is fine to leave it as .)
Write an expression for the total energy used in April
The problem says the plant has already used 72,000 kWh in the first 14 days and will use more during the rest of April.
So the total energy used in April is
Decide how to compare total usage with the allowed maximum
To meet the goal of at least a 15% reduction, the plant must not go over the maximum allowed usage. That means the total usage must be less than or equal to the allowed maximum.
In symbols, compare the total usage expression from Step 2 to the maximum allowed usage from Step 1 using the sign. This produces the inequality that matches the correct answer choice: