Question

which of the following inequalities is graphed on the coordinate plane?

a. ( y > 2x + 2 )

b. ( y geq 2x + 2 )

c. ( y leq 2x + 2 )

d. ( y < 2x + 2 )

Explanation:

Step1: Analyze the line type

The graph has a solid red line. A solid line in an inequality graph means the inequality includes equality (i.e., $\geq$ or $\leq$), while a dashed line would mean $>$ or $<$. So we can eliminate options A ($y > 2x + 2$) and D ($y < 2x + 2$) since they have dashed - line - equivalent inequalities.

Step2: Analyze the shaded region

To determine if it's $\geq$ or $\leq$, we look at the shaded region. Let's take a test point not on the line. The line is $y = 2x+2$. Let's use the origin $(0,0)$? Wait, no, the origin is on the line? Wait, the line passes through $(0,2)$ and let's find another point. The slope is 2 (since from $(0,2)$ to $(1,4)$ the rise is 2 and run is 1). Wait, let's take a point in the shaded region. The shaded region is above the line? Wait, no, looking at the graph, the shaded area is to the left of the line? Wait, no, let's check the inequality direction. Let's pick a point in the shaded region. Let's take $x=-3$, $y = - 4$. Plug into $y$ and $2x + 2$: $2(-3)+2=-6 + 2=-4$. So $y=-4$ and $2x + 2=-4$. Wait, no, maybe another point. Wait, the line is $y = 2x+2$. Let's take a point below the line, say $(0,0)$. Plug into $y$ and $2x + 2$: $0$ vs $2(0)+2 = 2$. So $0<2$. But the shaded region is not where $(0,0)$ is. Wait, the shaded region is above the line? Wait, the line goes through $(0,2)$ and $(-1,0)$ (since when $y = 0$, $0=2x + 2\Rightarrow x=-1$). The shaded area is on the side where, for example, when $x = 0$, the shaded area is above $y = 2$? Wait, no, the graph shows the shaded area is to the left of the line. Wait, maybe I made a mistake. Wait, the line is $y=2x + 2$. Let's check the inequality for the shaded region. Let's take a point in the shaded region, say $(-2,-2)$. Plug into $y$ and $2x+2$: $y=-2$, $2x + 2=2(-2)+2=-4 + 2=-2$. So $y=-2$ and $2x + 2=-2$, so $y = 2x+2$ at that point. Wait, no, another point: $(-3,-4)$. $2x+2=2(-3)+2=-4$, so $y=-4$ and $2x + 2=-4$, so $y=2x + 2$. Wait, maybe the shaded region is where $y\leq2x + 2$? Wait, no, let's check the line type. The line is solid, so it's either $\geq$ or $\leq$. Let's see the direction of the inequality. If the shaded region is below the line, then $y\leq2x + 2$. Wait, when $x = 0$, the line is at $y = 2$. The shaded region at $x = 0$ is below $y = 2$? Wait, the graph shows the shaded area is on the left - hand side of the line. Wait, maybe I should re - evaluate. The general form of a linear inequality: if the line is $y=mx + b$, and the shaded region is above the line, then $y\geq mx + b$ (if solid line) or $y>mx + b$ (if dashed). If below, then $y\leq mx + b$ (solid) or $y

Answer:

B. $y\geq2x + 2$