Question

of a system of inequalities is shown. which system is represented by the graph? $\

$$\begin{cases} y > 2x \\\\ x + 2y \\leq -8 \\end{cases}$$

$ $\

$$\begin{cases} y < 2x \\\\ x + 2y \\geq -8 \\end{cases}$$

$ $\

$$\begin{cases} y \\geq 2x \\\\ x + 2y < -8 \\end{cases}$$

$

Explanation:

Step1: Analyze dashed line (y=2x)

The dashed line passes through (0,0) and (1,2), so its equation is $y=2x$. The shaded region is below this line, and since it's dashed, the inequality is $y < 2x$.

Step2: Analyze solid line

The solid line passes through (0,-4) and (-8,0). Find its equation:
Slope $m=\frac{-4-0}{0-(-8)}=-\frac{1}{2}$. Using slope-intercept form $y=mx+b$, $b=-4$, so $y=-\frac{1}{2}x-4$. Multiply by 2: $2y=-x-8$, rearranged to $x+2y=-8$. The shaded region is above this solid line, so the inequality is $x+2y \geq -8$.

Step3: Match to options

Combine the two inequalities: $y < 2x$ and $x+2y \geq -8$.

Answer:

$y < 2x$
$x + 2y \geq -8$