a student earns $11 per hour for tutoring and $9 per hour as a teachers aide. let x = the number of hours…

Explanation:

Step1: Identify the inequality for tutoring hours

The tutoring - center requires at least 3 hours and no more than 7 hours of tutoring per week. So the inequality for the number of tutoring hours $x$ is $3\leq x\leq7$.

Step2: Recall the total - hours inequality

We also have the inequality $x + y\leq18$, which represents the total number of hours the student can work between tutoring and being a teacher's aide.

Step3: Consider non - negativity

Since the number of hours cannot be negative, we also have $x\geq0$ and $y\geq0$.

The set of inequalities that describe the constraints are: $3\leq x\leq7$ $x + y\leq18$ $x\geq0$ $y\geq0$

If you want to graph these inequalities:

For $x\geq3$:

Draw a vertical line $x = 3$ and shade the region to the right of this line.

For $x\leq7$:

Draw a vertical line $x = 7$ and shade the region to the left of this line.

For $x + y\leq18$:

Rewrite it in slope - intercept form $y=-x + 18$. Plot the y - intercept at $(0,18)$ and the x - intercept at $(18,0)$. Draw a solid line (because of the $\leq$ sign) and shade the region below the line.

For $x\geq0$ and $y\geq0$:

We are restricted to the first quadrant.

Answer:

The system of inequalities is $\left{\begin{array}{l}3\leq x\leq7\x + y\leq18\x\geq0\y\geq0\end{array}\right.$