Download Set Operations and Compound Inequalities

Set Operations and Compound Inequalities
A compound inequality is two or more inequalities connected by the words “and” or “or”. An
example of a compound inequality is x + 3 < 1 and x − 4 > −12 .
Definition of intersection of two sets: The symbol for the intersection of A and B is A ∩ B . The
intersection of two sets is the set of elements they have in common; those elements in both set A
and B. Remember intersection means “and”.
Example 1: Find the intersection of sets A and B where A = {3, 4,5, 6} and B = {5, 6, 7} .
A ∩ B =_________________________
To solve a compound inequality with the word “and”
1. Solve each inequality individually.
2. The solution is all the numbers that satisfy both inequalities
(where the 2 graphs overlap).
Example 2: Solve the compound inequality: x + 3 < 1 and x − 4 > −12 . Write the solution in
interval notation and put the solution on a number line.
x +3 <1
x < −2
and
and
x − 4 > −12
x > −8
x < −2
x > −8
The Solution
Write the solution in interval notation ____________________
Write the solution in set-builder notation ______________________________
Example 3: Solve the compound inequality: 2 y ≤ 4 y + 8 and 3 y ≥ −9 . Write the solution in
interval notation and put the solution on a number.
solution
interval notation _____________________
Example 4: Solve: x + 2 > 3 and 2 x + 1 < −3 . Write the solution in interval notation and put it
on a number line.
solution
interval notation _____________________
Definition of the union of two sets: The symbol for the union of A and B is A ∪ B . The union
of two sets is the set of elements either in set A or in set B. Remember union means “or”. See
the diagram on page 107
Example 5: Find the union of sets A and B where A = {3, 4,5, 6} and B = {5, 6, 7} .
A ∪ B =_________________________
-----------------------------------------------------------------------------------------------------------------To solve a compound inequality with the word “or”
1. Solve each inequality individually.
2. The solution is all the numbers that satisfy either one of the inequalities.
(everything in both graphs).
Example 6: Solve the compound inequality: y − 1 > 2 or 3 y + 5 < 2 y + 6 . Write the solution in
interval notation and put it on a number line.
y −1 > 2
y>3
or
or
3y + 5 < 2 y + 6
y+5< 6
y<1
y>3
y <1
Solution
Write the solution in interval notation _______________________________
Warning: The solution was y > 3 or y < 1 , both branches. DO NOT write this as 3 < y < 1
because this says that 3 is less than 1 which is obviously not true. Also, DO NOT write is as
1 < y < 3 . It is true that 1 < 3 but this also says that y < 3 but that’s false because, for example, 2
was not a solution. We were given that y > 3. A non-continuous interval is written in two
separate parts every time.
Example 7: Solve x < 5 or x < −3 . Write the solution in interval notation and put it on a number
line.
solution
interval______________________
Example 8: Solve 3 x − 2 ≤ 13 or x + 5 ≥ 7 . Write the solution in interval notation and put it on a
number line.
solution
interval notation _________________
---------------------------------------------------------------------------------------------------------------Example 9: Graph the compound inequality (5,11] ∩ [6, ∞) . Express the set in the simplest
interval form.
solution
interval notation _________________
Go back through all of the problems and change each union to intersection and each intersection to union. How does it change the solutions?