Question

the dj plans to spend more than $45 but no more than $75 monthly to purchase music for his playlist. the subscription is $15 per month, and the cost of downloading each music selection is $3 per minute. write and solve a compound inequality representing s, the number of minutes that may be purchased in one month.

Explanation:

Step1: Define the variable

Let \( s \) be the number of minutes of music downloaded. The total cost is the monthly subscription plus the cost per minute times the number of minutes, so the total cost \( C = 15 + 3s \).

Step2: Set up the compound inequality

The DJ spends more than \$45 but no more than \$75, so \( 45 < 15 + 3s \leq 75 \).

Step3: Solve the left inequality

Subtract 15 from both sides: \( 45 - 15 < 3s \), which simplifies to \( 30 < 3s \). Then divide by 3: \( \frac{30}{3} < s \), so \( 10 < s \).

Step4: Solve the right inequality

Subtract 15 from both sides: \( 3s \leq 75 - 15 \), which simplifies to \( 3s \leq 60 \). Then divide by 3: \( s \leq \frac{60}{3} \), so \( s \leq 20 \).

Step5: Combine the inequalities

Combining the two results, we get \( 10 < s \leq 20 \).

Answer:

The compound inequality is \( 45 < 15 + 3s \leq 75 \) and the solution is \( 10 < s \leq 20 \), meaning the number of minutes \( s \) satisfies \( 10 < s \leq 20 \).