Graph the following system of inequalities

Question

Graph the following system of inequalities. Show (by shading in) the feasible region.

$$x+2y\leq 12$$ $$2x+y\leq 12$$

$$x\geq 0 , y \geq 0$$

I would like to know how to graph these inequalities.

Answer 1

first step : draw their lines second step : check a point in the inequality to find out left or right area

Answer 2

To graph such an inequality, start by graphing the boundary.

For example, the boundary of $x+ 2y\le 12$, is the line $x+ 2y= 12$. To graph a line, you only need two points. If x= 0, that equation becomes 2y= 12 so y= 6. Mark (0, 6) on the graph. If y= 0, that equation becomes x= 12 so mark (12, 0) on the graph. Draw the line through those two points.

That line, where x+ 2y= 12, separates $x+ 2y \lt 12$ from $x+ 2y \gt 12$. It is easy to see that (0, 0) satisfies $0+ 2(0)= 0 \lt 12$ so every point on the same side of that line as (0, 0) satisfies this inequality. Do the same for each of the other inequalities.

The "feasible region", the region where all of these inequalities are satisfied, is where those all overlap- their "intersection".