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 dashed line 1 equation

First, identify the orange dashed line: it has a y-intercept of 2 and slope 2. Its equation is $y=2x+2$. The shaded region is above this line, so the inequality is $y>2x+2$.

Step2: Find dashed line 2 equation

Next, identify the gray dashed line: it has a y-intercept of 1 and slope $-\frac{1}{2}$. 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 the point (-3,1):
For $y>2x+2$: $1>2(-3)+2 \implies 1>-4$, which is true.
For $y<-\frac{1}{2}x+1$: $1<-\frac{1}{2}(-3)+1 \implies 1<2.5$, which is true.

Answer:

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