Question

which point in the solution set of the given inequalities?

Answer 1

In the image, we can identify the region of the plane in which the two subregions overlap (the part of the graph which has vertical and horizontal lines). We can obtain the values of x and y in that region by means of the inequalities, in this way:

$\begin{cases}x+y>2 \\ 4x+y\ge-1\end{cases}$

We solve the first inequality for y

$y>2-x$

This gives us the first constrain for the value of y

Now, to obtain the second constrain we simply need to use the second inequality:

$\begin{gathered} 4x+y\ge-1 \\ \Rightarrow y\ge-1-4x \end{gathered}$

Finally, the last step is to get the constraints for the value of x.

For this, we can directly analyze the image of the region, notice how it covers any value of x. This means that there are no constraints for the value of x.

$\Rightarrow x\in\mathfrak{\Re }$

So, the region is given by:

$x\in\mathfrak{\Re },\begin{cases}y>2-x \\ y\ge-4x-1\end{cases}$

Finally, the point that is in the region is the one that satisfies the previous inequalities.

1. (2,0)

If x=2, then

$\begin{gathered} ,\begin{cases}y>2-2=0 \\ y\ge-4(2)-1=-9\end{cases} \\ \Rightarrow y>0 \end{gathered}$

So, (2,0) cannot be the correct option

2. (0,2)

$\begin{gathered} \begin{cases}y>2-0=2 \\ y\ge-4(0)-1=-1\end{cases} \\ \Rightarrow y>2 \end{gathered}$

This means that (0,2) cannot be the right option

3. (0,3) notice that the same condition (y>2) holds in this situation too (since x=0)

But y=3>2. This point indeed satisfies the conditions! (0,3) is the answer

4. (0,-1) Notice that in this case y=-1 but it should happen that y>2. So this point cannot be in the region