Download Section 9.2 - Compound Inequalities

Math 27 Section 9.2 - Part II - Page 1
Section 9.2 - Compound Inequalities - Part II
I.
Definitions
A.
A set is a collection of objects.
1.
Notation - {1, 2, 3} is read "the set of elements 1, 2, and 3".
2.
The members of the set are called the elements of the set.
3.
2 ∈ {1, 2, 3} is read "2 is an element of the set 1, 2, 3".
4.
5 ∉ {1, 2, 3} is read "5 is not an element of the set 1, 2, 3".
B.
The union of two sets is the combination of all the elements in both sets.
1.
Let A = {1, 2, 3} and B = {2, 4, 6}
The union of sets A & B (written A ∪ B) is
A ∪ B = {1, 2, 3, 4, 6}
2.
Now you try one: Let A = {3, 5, 8} and B = ∅ . Find A ∪ B.
Answer:
A ∪ B = {3, 5, 8}
II.
Properties of Inequalities - (Let a, b, and c be real numbers)
A.
Addition (Subtraction) Property
If a < b, then a + c < b + c
B.
III.
Writing Solutions
A.
Graphically
1.
If you do not have equality, use a parentheses.
2.
If you do have equality, use a square bracket.
B.
IV.
Multiplication (Division) Property
1.
If a < b and c > 0, then ac < bc.
2.
If a < b and c < 0, then ac > bc.
Interval Notation
1.
If you do not have equality, use a parentheses.
2.
If you do have equality, use a square bracket.
Compound Inequalities
A.
A compound inequality is formed when two or more inequalities are joined by the
words “and” or “or”.
B.
The solution of a compound inequality is dependent on the type of inequality.
1.
If we have an “and” inequality, the solution is the intersection of the solution
sets.
2.
If we have an “or” inequality, the solution is the union of the solution sets.
C.
Solving Compound Inequalities
1.
Solve and graph each inequality separately.
2.
Determine the type of inequality and graph the solution set accordingly.
3.
Write the interval corresponding to the solution set.
D.
Examples - Solve, graph, and write the interval.
4.
2x - 3 < 5 or 2x - 8 > 4
Obviously, this is an “or” inequality. We can never put this together as a
single inequality, it has to stay separate. So we need to solve each one
independently. We will do that by first adding 3 to both sides of the the first;
adding 8 to both sides of the second:
2x < 8 or 2x > 12
© Copyright 2008 by John Fetcho. All rights reserved.
Math 27 Section 9.2 - Part II - Page 2
Now divide everything by 2:
Answer:
x < 4 or x > 6
Since this is an “or” inequality, the solution set (graphically) will be the union
of the two solution sets:
[
0 2 4 6
]
Thus the solution set is everything on the number line except the numbers
between 4 and 6. So graphically, this will look like:
Answer:
]
[
4
6
The interval will be:
Answer:
( −∞, 4 ] ∪ [6, ∞ )
5.
Now you try one: -x + 6 > -3 or 4x - 2 > 12
Answer:
All real numbers
(− ∞, ∞ )
6.
We are becoming weighter adults. The average weight of U.S. women, ages
20 – 74, has jumped 24 pounds over four decades, while average height has
increased from 5-foot-3 to 5-foot-4. For U.S. men, the average weight has
increased 25 pounds, while the average height has increased from 5-foot-8 to
5-foot-9. The bar graph at the top of the right-hand column, page 625, shows
the average weight of U.S. men and women, ages 20-74, for five selected
years over four decades. (Page 625, #72)
Looking at the graph on page 625, we want to see which years meet the
criteria set forth for both sets in this problem. I’m going to call the sets A & B.
A = {x|x is a year for which women’s weight > 144}
B = {x|x is a year for which men’s weight < 182}
So looking at the bar graph, we see that:
A = {1974, 1980, 1994, 2002}
B = {1962, 1974, 1980}
We want A ∪ B
Answer: A ∪ B = {1962, 1974, 1980, 1994, 2002}
© Copyright 2008 by John Fetcho. All rights reserved.