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: Set up the compound inequality

The total cost is the sum of the subscription cost (\(15\)) and the cost of downloading music (\(3s\)). The DJ spends more than \(45\) but no more than \(75\). So the inequality is \(45 < 15 + 3s\leq75\).

Step2: Solve the left - hand side of the inequality

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

Step3: Solve the right - hand side of the inequality

From \(15 + 3s\leq75\), subtract \(15\): \(15 + 3s-15\leq75 - 15\), which gives \(3s\leq60\). Divide by \(3\): \(\frac{3s}{3}\leq\frac{60}{3}\), so \(s\leq20\).

Answer:

The compound inequality is \(10 < s\leq20\). The graph that represents this is the one where there is an open circle at \(10\) (since \(s>10\)) and a closed circle at \(20\) (since \(s\leq20\)). So the correct graph is the third option (the one with an open circle at \(10\) and a closed circle at \(20\)).