A compound inequality is two inequalities joined together with the word and or the word or . The answer to a compound inequality contains the numbers which satisfy both inequalities. The solution will be the “intersection” between the sets of answers of the two inequalities. The intersection of two sets is all the elements that are in common between the sets. For example the sets (2, ∞) and (-∞, 8) have the numbers 2 to 8 in common, not including 2 and 8. In set notation, the intersection is (2, 8). Intersection between sets is noted with the symbol “∩” between the two sets.
Examples of compound inequalities containing the word and are y > 6 and y -9. Solutions to compound inequalities are generally written in interval notation, (such as (3, 8) ) or represented on a number line.
Example: Solve 2x + 6 ≤ 8 and 5(x – 2) > -20.
Solution:
We start by solving each inequality separately.
2x + 6 ≤ 8 and 5(x – 2 ) > -20
2x ≤ 2, and 5x – 10 > -20
x ≤ 1, and 5x > -10
x > -2
In set notation we have (-∞ , 1] ∩ (-2 , ∞), which is the set (-2 , 1]. If you have trouble obtaining the final answer, you can start by writing out numbers in each set and see what numbers are contained in both sets. When doing that you’ll see that all the numbers between -2 and 1, not including -2, are present in both sets. Note that intervals are said to be open if neither endpoints are contained in the solution (a , b). A half open interval is open on one end and closed on another (a, b] or [a, b). A closed interval contains both endpoints [a, b].
Example: Solve 6x – 8 ≥ 16 and 3x + 6 > 2x + 7.
Solution:
Solve each inequality separately.
6x – 8 ≥ 16 and 3x + 6 > 2x + 7
6x ≥ 24, and 3x > 2x + 1
x ≥ 4, and x > 1
In set notation we have [4, ∞) ∩ (1, ∞). The graph of the solution is the intersection of the two. In set notation the solution is [4, ∞). In some cases, there will be no solution. This happens where there are no numbers which make both parts of the compound inequality true.
Example: Solve 7x -6.
Solution:
Solve each inequality separately.
7x -6
x -2
In set notation we have (-∞ , -4) ∩ (-2 , ∞). It is clearly evident that there are no numbers which satisfy both inequalities.
Some compound inequalities contain the word or. The solution set for such inequalities are numbers that make either or both of the inequalities true. The solution is said to be the “union” of the two sets and in set notation is noted with the symbol “U”.
Example: Solve (2/3)x > 1 or (1/2)x
Solution:
Solve each inequality separately.
(2/3)x > 1 or (1/2)x
(3)(2/3)x > (3)(1) or 2(1/2)x
2x > 3 or x
x > 3/2
In set notation, we have (- ∞, -4) U (3/2, ∞).
Example: Solve x + 5 ≥ 3 or -x ≥ 7.
Solution:
Solve each inequality separately.
x + 5 ≥ 3 or -x ≥ -7
x ≥ -2 or x ≤ 7
In set notation we have [-2, ∞) U (- ∞, 7]. Notice that this solution is all real numbers and graphically the entire number line would be shaded. Inequalities might be written with 2 inequality symbols. Such inequalities are called double inequalities. An example of a double inequality is 2
This guide on compound inequalities should ease any confusion on the topic.
