Question

1. which one of the representations below corresponds to the solution set of the following system?

$\

$$\begin{cases} x + 2y \\leqslant 4 \\\\ y > \\frac{1}{2}(x+1)^2 - 2 \\end{cases}$$

$
a)
b)
c)
d)

Explanation:

Step1: Analyze linear inequality

Rewrite $x + 2y \leq 4$ to slope-intercept form:
$2y \leq -x + 4$
$y \leq -\frac{1}{2}x + 2$
This is a solid line (since $\leq$) with slope $-\frac{1}{2}$, y-intercept 2. The solution is below/on the line.

Step2: Analyze quadratic inequality

The inequality is $y > \frac{1}{2}(x+1)^2 - 2$. This is a dashed parabola (since $>$) opening upward, vertex at $(-1, -2)$. The solution is above the parabola.

Step3: Find overlapping region

The solution set is the area that is below the solid line and above the dashed parabola. Match this to the options.

Answer:

C) <corresponding graph with shaded region below solid line and above dashed parabola>