The solution set is the intersection of the solution sets of the simple inequalities.
OR Inequalities
The solution set is the union of the solution sets of the simple inequalities.
Let's consider them one at the time.
AND Inequalities
As stated in the table, the solution set of a compound inequality involving AND is the intersection of the solution sets of the simple inequalities. Therefore, if the intersection is empty, the compound inequality has no solution. Let's consider an example.
x < 2 AND x > 6
Let's now draw a graph showing the solution set of each simple inequality.
We can see above that there is no intersection, and thus the compound inequality has no solution.
OR Inequalities
As also stated in the table, the solution set of a compound inequality involving OR is the union of the solution sets of the simple inequalities. Therefore, if both simple inequalities have no solution, the compound inequality will not have solution. Let's consider an example.
x+2 < x+1 OR 2-x < 2-x
To find the solution of the compound inequality, we will solve each of the simple inequalities, one at the time. Let's start with the first one.
Since 2≮ 1, we have arrived at a false statement, and thus the first simple inequality has no solution. Let's now solve the second simple inequality.
Since 2≮ 2, we have again arrived at a false statement, and thus the second simple inequality has no solution.
Therefore, neither simple inequality has a solution. Thus, the compound OR inequality which is formed by them does not have a solution.
