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 \(5\).

Step2: Analyze the second inequality

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

This represents the region below and on the parabola opening downwards with vertex at \((0,0)\).

Step3: Identify the intersection region

The solution is the region inside the circle that lies below the parabola \(y = -x^2\).

Answer:

The correct graph is the one that shows the shaded region inside the circle of radius \(5\) centered at the origin, and below or on the parabola \(y = -x^2\) (opening downwards with its vertex at the origin).

(Note: The graph shown in the image is incorrect because it depicts a parabola opening upwards, \(y \ge x^2\), rather than \(y \le -x^2\).)