Let's break this down step-by-step to find the correct inequality and the minimum number of hours Julian needs to practice each remaining day:
1. Understand the given data:
- Julian has already practiced \(5 \frac{1}{3}\) hours this week.
- He wants to practice at least 7 hours in total for the week.
- He has 2 days left to reach his minimum practice goal.
2. Convert mixed number to improper fraction:
[tex]\[
5 \frac{1}{3} = 5 + \frac{1}{3} = \frac{15}{3} + \frac{1}{3} = \frac{16}{3}
\][/tex]
3. Calculate the remaining practice hours needed:
[tex]\[
\text{Minimum total hours needed} - \text{Hours already practiced} = 7 - 5 \frac{1}{3}
\][/tex]
[tex]\[
7 - \frac{16}{3} = \frac{21}{3} - \frac{16}{3} = \frac{5}{3} \approx 1.67 \text{ hours}
\][/tex]
4. Set up the inequality for the remaining 2 days:
Let \(x\) be the number of hours Julian needs to practice each of the next two days. Therefore, the total practice time for those 2 days is \(2x\).
The inequality representing the total practice time is:
[tex]\[
5 \frac{1}{3} + 2x \geq 7
\][/tex]
Let's rewrite it using the improper fraction:
[tex]\[
\frac{16}{3} + 2x \geq 7
\][/tex]
5. Solve the inequality:
[tex]\[
\frac{16}{3} + 2x \geq 7
\][/tex]
First, multiply both sides by 3 to clear the fraction:
[tex]\[
16 + 6x \geq 21
\][/tex]
Subtract 16 from both sides:
[tex]\[
6x \geq 5
\][/tex]
Divide by 6:
[tex]\[
x \geq \frac{5}{6} \approx 0.83 \text{ hours}
\][/tex]
So, Julian needs to practice at least \(\boxed{0.83}\) hours each of the next 2 days to meet his 7-hour weekly goal.
Therefore, the correct inequality from the given options is:
[tex]\[
5 \frac{1}{3} + 2x \geq 7
\][/tex]
