Question

which of the following graphs correctly depicts the system of inequalities below?
\\x^2 + y^2 \le 36\\
\\x^2 + y^2 > 16\\

Explanation:

Step1: Identify the boundary curves

The boundaries are circles centered at the origin:

$$x^2 + y^2 = 36 \implies r = 6$$

$$x^2 + y^2 = 16 \implies r = 4$$

Step2: Determine boundary line styles

The inequality \(x^2 + y^2 \le 36\) uses a solid boundary line.
The inequality \(x^2 + y^2 > 16\) uses a dashed boundary line.

Step3: Determine the shaded region

The solution set lies between the two circles:

$$16 < x^2 + y^2 \le 36$$

This represents the annular region (ring) between radius \(4\) and \(6\).

Answer:

The displayed graph correctly depicts the system of inequalities.