Question

the valley voices choir is practicing for a show. to feel prepared, they will practice for more than 43 hours. they also dont want to overuse their voices, so they will practice for less than 67 hours. so far, they have practiced a total of 11 hours. suppose they will decide to practice the same amount of time each week. let x be the number of hours per week they will practice for the remaining 8 weeks. (a) find the possible numbers of hours per week they will practice. write your answer as a compound inequality solved for x. (b) on the number line below, graph the solution that represents the possible numbers of hours per week they will practice.

Explanation:

Step1: Set up the compound - inequality

They have already practiced 11 hours. They will practice \(x\) hours per week for 8 weeks. The total number of hours of practice should be more than 43 and less than 67. So, the inequality is \(43<11 + 8x<67\).

Step2: Subtract 11 from all parts of the inequality

Subtract 11 from each part: \(43−11<11 + 8x−11<67−11\), which simplifies to \(32<8x<56\).

Step3: Divide all parts by 8

Divide each part of the inequality \(32<8x<56\) by 8: \(\frac{32}{8}<\frac{8x}{8}<\frac{56}{8}\), resulting in \(4 < x<7\).

Answer:

The compound - inequality is \(4 < x<7\). To graph it on a number line, draw an open circle at 4 and an open circle at 7, and shade the region between them.