The problem describes the constraints on a student's work hours and earnings at a college. Students can work no more than 20 hours per week and can earn a maximum of $320 per week. The jobs pay different rates, starting from $8.75 per hour. We need to write two inequalities representing these constraints and define the variables used.

1. Problem Description

The problem describes the constraints on a student's work hours and earnings at a college. Students can work no more than 20 hours per week and can earn a maximum of $320 per week. The jobs pay different rates, starting from$8.75 per hour. We need to write two inequalities representing these constraints and define the variables used.

2. Solution Steps

Let $h$ represent the number of hours a student works per week.

Let $e$ represent the amount a student earns per week.

The first constraint is that the student can work no more than 20 hours per week. This can be represented by the inequality:

$h \le 20$

The second constraint is that the student can earn a maximum of $320 per week. This can be represented by the inequality:

$e \le 320$

However, since the minimum rate is $8.75 per hour, the earnings$e$will be equal to the number of hours worked$h$multiplied by the hourly rate. Since we are considering the minimum rate, and we know that$e \le 320$, and since the rate is at least$8.75$, we can have the following inequality:

$8.75h \le e \le 320$. This implies that

$8.75h \le 320$

Dividing both sides by 8.75, we have:

$h \le \frac{320}{8.75}$

$h \le 36.57$

However, we are already given that $h \le 20$, which is stricter than $h \le 36.57$, so we use $h \le 20$.

The amount a student earns depends on the hourly rate multiplied by the number of hours worked. Let $r$ be the rate earned per hour.

Then $e = r*h$. Since the hourly rate is at least $8.75,$r \ge 8.75$.

Also, the earnings are at most $320, so$e \le 320$.

A valid inequality to represent the income is: $8.75h \le 320$. If we divide both sides by 8.75, we get:

$h \le \frac{320}{8.75} \approx 36.57$.

However, we also have the constraint that $h \le 20$. So a better inequality is:

$8.75h \le e$, and $e \le 320$. Therefore, we can say that $8.75h \le 320$.

Since $h \le 20$, the maximum income should be $20 \cdot 8.75 = 175$.

The two inequalities are $h \le 20$ and $e \le 320$.

Alternatively, we can also use $8.75h \le 320$, if the rate is fixed at $8.75 per hour. In reality, the student can have several jobs with different rates. Let$h_i$be the hours worked at job$i$and$r_i$be the rate for job$i$. The total number of hours$\sum h_i \le 20$and total earnings$\sum r_i h_i \le 320$. If the rates$r_i$were all$8.75$, then$\sum 8.75 h_i = 8.75 \sum h_i \le 320$.

Since $\sum h_i \le 20$, $8.75(20) \le 320$.

3. Final Answer

The two inequalities are:

$h \le 20$

$e \le 320$

where $h$ represents the number of hours a student works per week and $e$ represents the amount a student earns per week.