A compound inequality describes a quantity with both a lower and an upper limit, signaled here by “at least” and “at most.” Write the total cost as a fixed fee plus a per-minute charge, then place it between two percentages of the fixed fee. Subtract the fee before dividing by the per-minute rate; the extra cost is in dollars, not minutes.
Hints
- Hint 1
The fixed fee is paid once, while the call charge grows by for every minute. What expression adds the monthly texting fee to the cost of minutes?
- Hint 2
A percent base is the amount a percentage is taken of. Both percentages are of the texting fee, so use on both ends of an inequality and put the total cost in the middle.
- Hint 3
A compound inequality joins two comparisons in one chain. Subtract from all three parts, then divide by the positive charge per minute. That converts the extra cost in dollars into a range of minutes.
Step-by-step
Bound the total, then isolate the minutes
Step 1Write the total monthly cost
The dollars is paid once for texting. Voice calls cost dollars for each of minutes, so the total cost is . Don't multiply by ; the texting fee doesn't change with the minutes.
- Step 2
Put the total between its limits
The base, the amount each percent is taken of, is the texting fee . So both percentages apply to . “At least” includes the lower limit, and “at most” includes the upper limit: .
- Step 3
Separate the voice-call cost
Subtract from all three parts to remove the fixed fee: . The limits now describe extra dollars for calls, not minutes yet.
- Step 4
Convert the cost range to minutes
Divide all three parts by , the positive charge per minute, so the inequality signs stay the same: . Type and into Desmos. It shows and , so the allowed number of voice-call minutes satisfies . Choice B.