MHB How do you solve and plot inequalities with multiple variables?

[ai93]

I have the inequalities $$2<x<6,\quad 1<y<5,\quad y-2\le2x, \quad-2y\ge8-4x$$ I have to solve these and plot it in a graph and show the region where they satisfy. I understand you have to find the common area and shade it.

How do you find the points to plot for $$2<x<6\quad and\quad 1<y<5$$

I think I have solved $$y-2\le2x\quad and -2y\ge8-4x$$ to $$y\le2x+2\quad and \quad y\le-4+2x$$ I am just unsure on the first two. Will making a X and Y table help to find the points?

 

[Siron]

Well, the region $2<x<6$ contains all the points in the $x,y$-plane where the $x$-component lies between $2$ and $6$ (both not included).

 

[ai93]

Well, the region $2<x<6$ contains all the points in the $x,y$-plane where the $x$-component lies between $2$ and $6$ (both not included).

If i made a graph, would it go through the $y$ axis diagonal through the points $2$ and $6$?

 

[MarkFL]

If i made a graph, would it go through the $y$ axis diagonal through the points $2$ and $6$?

For the inequality $2<x<6$, I would begin by graphing the lines $x=2$ and $x=6$ with dashed lines since the inequality is strict on both sides of $x$. Now you have divided the plane into 3 regions. which of these regions should you shade?

 

[ai93]

For the inequality $2<x<6$, I would begin by graphing the lines $x=2$ and $x=6$ with dashed lines since the inequality is strict on both sides of $x$. Now you have divided the plane into 3 regions. which of these regions should you shade?

Because it is x>2 is greater than, I shade everything to the right. And becuase x<6 is less than, shade everything to the left? I will be left with a shaded region in between 2 and 6?

 

[MarkFL]

Because it is x>2 is greater than, I shade everything to the right. And becuase x<6 is less than, shade everything to the left? I will be left with a shaded region in between 2 and 6?

Yes, good! (Sun)

That's what $2<x<6$ means...any value of $x$ on the interval $(2,6)$, i.e., any value of $x$ in between 2 and 6, but not including 2 and 6. :D

So, the region you have shaded contains all the points in the plane for which the $x$-coordinate satisfies the given compound inequality.

 

[MarkFL]

Your graph should look like the following:

View attachment 3415