Question

which of the following graphs correctly depicts the system of inequalities below?

\\x^2 + y^2 \le 25\\
\\y + x^2 \le 0\\

Explanation:

Step1: Analyze the first inequality

$$x^2 + y^2 \le 25$$

This represents the region inside and on a circle centered at \((0,0)\) with radius \(5\), drawn with a solid boundary.

Step2: Analyze the second inequality

$$y + x^2 \le 0 \implies y \le -x^2$$

This represents the region on and below the parabola \(y = -x^2\), opening downwards with vertex at \((0,0)\), drawn with a solid boundary.

Step3: Identify the intersection region

The solution is the overlapping region inside the circle and below the parabola, which is correctly depicted by the darker shaded region in the provided graph.

Answer:

The correct graph is the one shown in the image, which features a solid boundary for the circle \(x^2 + y^2 = 25\) shaded inside, and a solid boundary for the parabola \(y = -x^2\) shaded below, with their intersection shaded in a darker overlapping region.