Question

identifying a solution from the graph of a system of two-variable inequalities
which system of inequalities with a solution point is represented by the graph?
○ $y > 2x - 2$ and $y < -\frac{1}{2}x - 1$; $(3, 1)$
○ $y > 2x - 2$ and $y < -\frac{1}{2}x + 1$; $(-3, 1)$
○ $y > 2x + 2$ and $y < -\frac{1}{2}x - 1$; $(3, 1)$
○ $y > 2x + 2$ and $y < -\frac{1}{2}x + 1$; $(-3, 1)$

Explanation:

Step1: Find orange line equation

The orange dashed line has slope $2$ and y-intercept $-2$, so its equation is $y=2x-2$. The shaded region is above this line, so the inequality is $y>2x-2$.

Step2: Find blue line equation

The blue dashed line has slope $-\frac{1}{2}$ and y-intercept $1$, so its equation is $y=-\frac{1}{2}x+1$. The shaded region is below this line, so the inequality is $y<-\frac{1}{2}x+1$.

Step3: Verify solution point

Check $(-3,1)$ in both inequalities:
For $y>2x-2$: $1>2(-3)-2=-8$, which is true.
For $y<-\frac{1}{2}x+1$: $1<-\frac{1}{2}(-3)+1=\frac{3}{2}+1=\frac{5}{2}$, which is true.

Answer:

B. $y > 2x - 2$ and $y < -\frac{1}{2}x + 1; (-3, 1)$