Remember that the points that are solutions to the system of linear inequalities satisfy both inequalities. Review what information the inequality symbols provide.
See solution.
Practice makes perfect
There are different ways to decide which of the regions represent the solution set of a system of linear inequalities. We will review how to do this using test points and by checking the inequalities signs.
Using a test point
If substituting the point makes both inequalities hold true, then it means that the point is a solution to the system, and the region were it lies would be the one we need to shade. Let's see an example.
System of linear inequalities
y≤ x+1 y ≤ -x +2 1.25cm
In this case we chose (0,0) as our test point for simplicity. We just need to make sure that our chosen test point does not lie on either of the boundary lines, as that would not give us enough information. Let's substitute our test point in the first inequality.
The point makes both inequalities true. Therefore, it is part of the solution set of the system and we can shade the region containing it.
Remember that when the lines are solid (≤ or ≥) the boundary line is part of the solution set as well. If they are dashed, then the boundary line is not included in the solution set.
However, notice that if the test point does not satisfy both inequalities and even if it satisfies just one of them, we may need to use more test points.
By checking the signs
If our inequalities are written in slope-intercept form, we can identify which half-plane to shade for each individual inequality by just analyzing the inequality symbol. If the symbol used is > or ≥, we would shade the region above the boundary line.
If the symbol used is either < or ≤, we should shade below the boundary line.
Then, we know how to shade the corresponding region for each individual inequality. Let's try with the same system as before.
Now, remember that the solution set for a system of linear inequalities is the overlapping region of the individual inequalities (and corresponding parts of the boundary lines for ≥ or ≤ signs).
