Question

The graph below represents which system of inequalities?graph of two infinite lines that intersect at a point. One line is solid and goes through the points 0, 2, negative 2, 4 and is ay ≤ −2x + 3y ≤ x + 3 by ≥ −2x + 3y ≥ x + 3 cy ≤ −3x + 2y ≤ −x + 2 dy > −2x + 3y > x + 3

Answer 1

Notice that both lines are solid lines, and in both regions, the solution set is below the line.

Now, to answer this question first we will compute the equation of both lines.

Recall that the equation of a line that passes through two given points is given by the following formula:

`y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)\text{.}`

From the given graph we get that one line passes through the points (2,0) and (0,2), and the other one passes through the points (0,2) and (1,-1), therefore, its equations are:

`\begin{gathered} y-0=(2-0)/(0-2)(x-2), \\ y-2=(-1-2)/(1-0)(x-0)\text{.} \end{gathered}`

Simplifying the above equations we get:

`\begin{gathered} y=-x+2, \\ y-2=-3x\rightarrow y=-3x+2. \end{gathered}`

Since the regions are below the lines, and the lines are solid lines, then the system of inequalities is:

`\begin{gathered} y\le-x+2, \\ y\le-3x+2. \end{gathered}`

Answer: Option C.