Question

Which of the following systems linear inequalities does the graph represent? Select all that apply.

Answer 1

To answer this question, first of all, we need to find the equation for each of the lines on the graph. To do so, we have to identify two points for each line, and then use the two-point form equation (of the line) to find the equation of each line.

Then, we have:

Equation of the solid line

We identify two points: (1, 1) and (2, 2).

Then we have:

`y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)`

Then if we label both points above:

• (1, 1) ---> x1 = 1, y1 = 1

,

• (2, 2) ---> x2 = 2, y2 = 2

Therefore:

`\begin{gathered} y-1=(2-1)/(2-1)(x-1) \\ y-1=1(x-1)=(x-1) \\ y-1=x-1\Rightarrow y=x-1+1\Rightarrow y=x \end{gathered}`

Therefore, the equation of the line for the solid line is y = x.

Equation of the dotted line

We can proceed similarly as we did before. Then we have:

The two points are (2, 4) and (4, 8).

Then we can label them as follows:

• (2, 4) ---> x1 = 2, y1 = 4

,

• (4, 8) ---> x2 = 4, y2 = 8

`\begin{gathered} y-4_{}=\frac{8-4_{}}{4_{}-2_{}}(x-2) \\ y-4=(4)/(2)(x-2) \\ y-4=2(x-2)\Rightarrow y-4=2x-4 \\ y=2x-4+4 \\ y=2x \end{gathered}`

Therefore, the equation for the dotted line is y = 2x.

We need to remember that when we have the symbols < or > the inequality is represented by a dotted line. Conversely, when it appears additionally an equal sign in the inequality symbol, ≥ or ≤ the line is represented as a solid line.

If we see the graph, we can say that the graph is represented by the following inequalities:

`\begin{gathered} y>2x\Rightarrow Dotted\text{ line} \\ y\ge x\Rightarrow Solid\text{ line} \end{gathered}`

Since if we represent them graphically, we have - using a graphing calculator:

We can observe that the shaded area is the same as the one represented in the graph given in the question.

We can prove that if we take a point in the solution region as follows:

1. For example, we can test the point (-2, 2). Then we have:

`\begin{gathered} 2>2(-2)\Rightarrow2>-4\Rightarrow True \\ 2\ge-2\Rightarrow2>-2\Rightarrow True \end{gathered}`

Now, using this we can check if the other system of inequalities is equivalent to this:

We can check the following inequalities:

`\begin{gathered} -2x+y>0\Rightarrow y>2x \\ -x+y\ge0\Rightarrow y\ge x \end{gathered}`

Which is equivalent to the previous one.

In summary, we have that the systems of linear inequalities are the following:

`\begin{cases}-2x+y>0 \\ -x+y\ge0\end{cases}`

[Third option]

And

`\begin{cases}y>2x \\ y\ge x\end{cases}`

[Fourth option]

[The second and the third options are not answers since both have the symbol, ≥, - and it means that we have two solid lines representing the inequality system.]

[In the first option we have that the first inequality represents a solid line, and this is not the case. The dotted line is represented by y > 2x.]