a graphed system of linear inequalities is shown below.
which of the following ordered pairs is a solution to the system of inequalities? select all that apply.
a. (0, -3)
b. (-4,1)
c. (-3, -4)
d. (5, -2)
e. (0,1)

Question

a graphed system of linear inequalities is shown below.
which of the following ordered pairs is a solution to the system of inequalities? select all that apply.
a.  (0, -3)
b.  (-4,1)
c.  (-3, -4)
d.  (5, -2)
e.  (0,1)

Accepted Answer

To determine which ordered pairs are solutions to the system of linear inequalities, we check if each point lies within the shaded region (the solution set) of the graphed system.

### Analyzing Each Option:
- **Option A: $(0, -3)$**
  Locate $(0, -3)$ on the graph. The point $(0, -3)$ lies on the solid line (or within the shaded region? Wait, looking at the graph, the shaded region is the overlapping area. Let's re - evaluate. Wait, the shaded region is the area that is above the dashed line and below the solid line (or vice - versa? Wait, the shaded region is the area that is in the intersection of the two inequality regions. Let's check the position of $(0, -3)$. Wait, the y - coordinate is - 3. Looking at the graph, the shaded region at $x = 0$ has $y$ values above a certain line. Wait, maybe a better way: a solution to the system of inequalities must lie in the shaded region. Let's check each point:
  - For $(0,-3)$: Looking at the graph, the point $(0, - 3)$ is on the boundary of the shaded region? Wait, no. Wait, the shaded region is the area that is, for example, above the dashed line and below the solid line? Wait, the dashed line has a positive slope, and the solid line has a negative slope. The shaded region is the area that is to the right of the dashed line (since the dashed line goes from $(-10,-5)$ to $(10,6)$ approximately) and to the right of the solid line (which goes from $(-10,10)$ to $(0, - 10)$)? Wait, no. Let's use the method of substituting the points into the inequality (even though we can't see the exact inequalities, we can use the shaded region). The shaded region is the area where the two inequalities overlap. Let's check each point:
  - $(0,-3)$: Is this in the shaded region? Looking at the graph, the shaded region at $x = 0$ has $y$ values from, say, - 10 (the bottom of the solid line) up to 6 (the top of the dashed line). Wait, no. Wait, the solid line passes through $(0,-10)$ and has a slope of $\frac{10 - (-10)}{-10-0}=\frac{20}{-10}=-2$, so its equation is $y=-2x - 10$? Wait, no, when $x = 0$, $y=-10$, when $x = 5$, $y=-2\times5-10=-20$? No, that's not matching. Wait, maybe the solid line has a slope of $m=\frac{0 - (-10)}{10 - 0}=1$? No, the solid line goes from the top left to the bottom right. Let's take two points on the solid line: when $x = 0$, $y=-10$; when $x = 5$, $y = 0$? Wait, no, the solid line intersects the $y$ - axis at $(0,-10)$ and the $x$ - axis at $(5,0)$? Wait, slope $m=\frac{0-(-10)}{5 - 0}=2$? No, that's not right. Alternatively, maybe the two lines are $y=\frac{1}{2}x + 1$ (dashed) and $y=-2x-10$ (solid). But maybe it's easier to use the shaded region. The shaded region is the area that is above the dashed line and below the solid line? Wait, no. Let's check the points:
  - **Option D: $(5,-2)$**
    The point $(5,-2)$: $x = 5$, $y=-2$. Looking at the graph, this point is within the shaded region.
  - **Option E: $(0,1)$**
    The point $(0,1)$: $x = 0$, $y = 1$. This point is within the shaded region.
  - **Option A: $(0,-3)$**
    The point $(0,-3)$: $x = 0$, $y=-3$. This point is below the shaded region (since the shaded region at $x = 0$ starts from $y$ values above, say, - 10 but the lower boundary at $x = 0$ is the solid line? Wait, no, maybe I made a mistake. Wait, let's re - examine the graph. The shaded region is the area that is to the right of the dashed line (which has a positive slope) and to the right of the solid line (which has a negative slope). Let's check the $x$ and $y$ coordinates:
    - For a point to be in the shaded region, it must satisfy both inequalities. Let's take point D: $(5,-2)$. $x = 5$ (positive), $y=-2$. It's in the shaded area.
    - Point E: $(0,1)$. $x = 0$, $y = 1$. It's in the shaded area.
    - Point A: $(0,-3)$: $y=-3$ is below the shaded region (the shaded region at $x = 0$ has $y$ values from, say, - 10 up to 6, but the point $(0,-3)$ is on the solid line? Wait, no, the solid line at $x = 0$ is $y=-10$? No, I think I messed up. Let's use the fact that the solution to a system of inequalities is the set of points in the intersection of the solution sets of each inequality. The shaded region is the intersection. So we check if the point is in the shaded region:
      - **Point A: $(0,-3)$**
        Is $(0,-3)$ in the shaded region? Looking at the graph, the shaded region does not include $(0,-3)$ (it's below the shaded area).
      - **Point B: $(-4,1)$**
        $x=-4$ (negative), $y = 1$. This point is to the left of the shaded region (since the shaded region starts at $x$ values greater than, say, - 5 or so).
      - **Point C: $(-3,-4)$**
        $x=-3$ (negative), $y=-4$. This point is to the left and below the shaded region.
      - **Point D: $(5,-2)$**
        $x = 5$ (positive), $y=-2$. This point is in the shaded region.
      - **Point E: $(0,1)$**
        $x = 0$, $y = 1$. This point is in the shaded region.

# Brief Explanations:
To determine the solutions, we check if each ordered pair lies within the shaded region (the solution set of the system of linear inequalities).
- **A. $(0, - 3)$:** Not in the shaded region.
- **B. $(-4, 1)$:** Not in the shaded region (left of the shaded area).
- **C. $(-3, - 4)$:** Not in the shaded region (left and below the shaded area).
- **D. $(5, - 2)$:** Lies within the shaded region.
- **E. $(0, 1)$:** Lies within the shaded region.

# Answer:
D. $(5, - 2)$, E. $(0, 1)$

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Analyzing Each Option:

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