Compound Inequalities: Union and Intersection of Inequalities

Union of Inequalities

When we see a statement like "x < 7 or x≥11", written in set notation as {x : x < 7orx≥11}, the word or denotes the union of the two sets of numbers which satisfy each inequality. Thus, {x : x < 7orx≥11} = {x : x < 7}∪{x : x≥11}. This is the set of values which satisfy eitherx < 7orx≥11. The value 5 satisfies the statement, as does the value 14.

We can graph the union of two inequalities on the number line. To do this, simply graph both inequalities:

Every point on the dark line is a member of the set {x : x < 7orx≥11}.

Sometimes the two inequalities will overlap. This is fine. The set of all values which satisfy either inequality is the set of all points which satisfy one or the other or both--this includes the overlap.

Intersection of Inequalities

When we see a statement like "0≤x < 4", also written as "0≤x and x < 4", or as {x : 0≤x < 4}, the compound inequality or the word and denotes the intersection of the two sets of numbers which satisfy each inequality. Thus, {x : 0≤x < 4} = {x : 0≤x}∩{x : x < 4}. This is the set of values which satisfy both0≤xandx < 4. The value 2 satisfies the statement, but the value -3 does not, and the value 5 does not.

We can graph the intersection of two inequalities on the number line. To do this, lightly graph each inequality. Then darken the line which appears in the graph of both inequalities. Finally, erase the light line which does not appear in the graph of both inequalities: