Question

the tonal eclipse choir is practicing for a show. to feel prepared, they will practice for at least 23 hours. they also dont want to overuse their voices, so they will practice for at most 48 hours. so far, they have practiced a total of 13 hours. let x be the number of hours per week they will practice for the remaining 5 weeks. (suppose they will decide to practice the same amount of time each week.)
(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 total practice inequality

Let total practice time be $13 + 5x$. The total must be at least 23 hours:
$$13 + 5x \geq 23$$

Step2: Solve lower bound for x

Subtract 13, then divide by 5:
$$5x \geq 23 - 13$$
$$5x \geq 10$$
$$x \geq 2$$

Step3: Set up upper limit inequality

Total practice time is at most 48 hours:
$$13 + 5x \leq 48$$

Step4: Solve upper bound for x

Subtract 13, then divide by 5:
$$5x \leq 48 - 13$$
$$5x \leq 35$$
$$x \leq 7$$

Step5: Combine into compound inequality

Combine the two bounds for $x$.

Answer:

(a)

$2 \leq x \leq 7$

(b)

To graph this on a number line:

1. Draw a closed circle at $x=2$ (since the value is included in the solution) and a closed circle at $x=7$.
2. Shade the segment of the number line between these two closed circles to represent all values of $x$ from 2 to 7, inclusive.