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 cost function

The total cost \( C \) is the sum of the monthly subscription and the cost of the music minutes. The subscription is $15, and each minute of music is $0.3. So \( C = 15 + 0.3s \), where \( s \) is the number of minutes.

Step2: Set up the inequality

DJ wants to spend more than $45 but no more than $75. So \( 45 < 15 + 0.3s \leq 75 \).

Step3: Solve the left inequality

Subtract 15 from both sides: \( 45 - 15 < 0.3s \), which simplifies to \( 30 < 0.3s \). Then divide both sides by 0.3: \( \frac{30}{0.3} < s \), so \( 100 < s \).

Step4: Solve the right inequality

Subtract 15 from both sides: \( 0.3s \leq 75 - 15 \), which simplifies to \( 0.3s \leq 60 \). Then divide both sides by 0.3: \( s \leq \frac{60}{0.3} \), so \( s \leq 200 \).

Answer:

The compound inequality is \( 100 < s \leq 200 \), meaning the number of minutes \( s \) satisfies \( 100 < s \leq 200 \).