Question

y + 7 ≤ x²\
y < -x² + 5

Explanation:

Step1: Analyze the first inequality

We have the inequality \( y + 7\leq x^{2} \). To solve for \( y \), we subtract 7 from both sides of the inequality.
\( y\leq x^{2}-7 \)

Step2: Analyze the second inequality

The second inequality is already in the form solved for \( y \): \( y < -x^{2}+5 \)

Step3: Find the solution region (conceptual)

To find the solution to the system of inequalities, we need to find the region of the coordinate plane that satisfies both \( y\leq x^{2}-7 \) and \( y < -x^{2}+5 \) simultaneously. This would involve graphing both parabolas (the first is a parabola opening upwards with vertex at \( (0, - 7) \) and the second is a parabola opening downwards with vertex at \( (0,5) \)) and then identifying the area that is below (or on) the first parabola and below the second parabola.

If we were to find the points of intersection (to better understand the region), we set \( x^{2}-7=-x^{2}+5 \).

Step4: Solve for intersection

\( x^{2}+x^{2}=5 + 7\)
\(2x^{2}=12\)
\(x^{2}=6\)
\(x=\pm\sqrt{6}\approx\pm2.45\)

Substituting \( x = \sqrt{6} \) into \( y=-x^{2}+5 \) (we could use either equation), we get \( y=-6 + 5=-1 \). So the points of intersection are \( (\sqrt{6},-1) \) and \( (-\sqrt{6},-1) \). The solution region is the area that is below the downward - opening parabola \( y=-x^{2}+5 \) and below (or on) the upward - opening parabola \( y = x^{2}-7 \) between \( x=-\sqrt{6} \) and \( x=\sqrt{6} \) (and also considering the regions outside, but since the parabola \( y=-x^{2}+5 \) opens downward and \( y=x^{2}-7 \) opens upward, for \(|x|>\sqrt{6}\), \(x^{2}-7>-x^{2}+5\) so there is no solution in those regions as \( y \) can't be both less than \( -x^{2}+5 \) and less than or equal to \( x^{2}-7 \) when \( x^{2}-7>-x^{2}+5 \)).

Answer:

The system of inequalities is \(

$$\begin{cases}y\leq x^{2}-7\\y < -x^{2}+5\end{cases}$$

\), and the solution region is the area between \( x =-\sqrt{6} \) and \( x=\sqrt{6} \) that is below the parabola \( y=-x^{2}+5 \) and below (or on) the parabola \( y=x^{2}-7 \). If we were to present the solution in terms of \( y \) in terms of \( x \), the solution for \( y \) is \( y<\min(-x^{2}+5,x^{2}-7) \) (with the understanding of the domain from the intersection points).