Which of the following points lie in the solution set to the following system of inequalities?
y < −3x + 3
y < x + 2
To find the points that lie in the solution set to this system of inequalities, we need to find the overlapping region between the two individual solution sets.
First, let's graph each individual inequality:
y < −3x + 3
This inequality represents a line with a slope of -3 and a y-intercept of 3. It will be a dashed line (since it's a strict inequality) and we'll shade the region below the line to represent the solution set.
y < x + 2
This inequality represents a line with a slope of 1 and a y-intercept of 2. It will also be a dashed line (since it's a strict inequality) and we'll shade the region below the line to represent the solution set.
Now we need to find the overlapping region between these two shaded regions:
[asy]
import graph;
defaultpen(linewidth(0.7));
// y < -3x + 3
draw((-2,5)--(1,-6),dashed,Arrows);
fill((8/5,2/5)--(-2,5)--(1,2)--cycle,gray(0.7));
// y < x + 2
draw((-3,-1)--(4,6),dashed,Arrows);
fill((-3,-1)--(8/5,2/5)--(4,6)--cycle,gray(0.7));
// axes and labels
xaxis(-3,4,Arrows(4));
yaxis(-6,5,Arrows(4));
label("$x$",(4,0),E);
label("$y$",(0,5),N);
// points to check
dot("$(0,1)$",(0,1),NE);
dot("$(-1,-2)$",(-1,-2),SW);
dot("$(1,0)$",(1,0),N);
dot("$(1,2)$",(1,2),N);
[/asy]
From the graph, we can see that the overlapping region is the triangle bounded by the lines y = -3x + 3, y = x + 2, and the x-axis. Any point within this triangle will satisfy both inequalities, so the points in the solution set are:
$\boxed{(0,1), \ (-1,-2), \ (1,0), \text{ and } (1,2)}$