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Sets and Compound Inequalities
Introduction to Sets
A set is a collection of objects which are called elements. The
elements are distinct and, thus, are listed only once. One way to
identify a set is by the List or Roster Method, whereby the
elements are listed within braces {}. To indicate that an object is
an element of a set, the symbol ∈ is used. To indicate that an
object is not an element of a set, the symbol ∉ is used.
Example 1:
Write the first four months of a year as set M using the List Method,
identify one of the months as an element of the set, and identify a month
that is not an element of the set.
M = {January, February, March, April}
February ∈ M
July ∉ M
A set that has no elements is called an empty set and is denoted
by the symbol Ø.
Sets can be joined together using the intersection of sets or the
union of sets.
The intersection of two sets A and B is the set of all elements
that are common to both A and B and is denoted as
A∩B
The union of two sets A and B is the set of all elements in both
A and/or B and is denoted as
A∪B
Example 2:
Find the intersection: {2, 3, 4, 5, 6, 7} ∩ {1, 2, 5, 7, 8, 9}
{2, 5, 7}
Example 3:
Find the union: {2, 4, 6, 8} ∪ {3, 5, 7}
{2, 3, 4, 5, 6, 7, 8}
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Compound Inequalities
Similar to sets, several inequalities can be combined together to
form what are called compound inequalities.
One type of compound inequality is an OR inequality. This type
of inequality will result in a true statement from either one
inequality OR the other inequality OR both. This compound
inequality is called a disjunction.
The solution set of a disjunction is the union of the solution sets
of the individual inequalities.
A convenient way to graph a disjunction is to graph each
individual inequality above the number line, then move them both
onto the actual number line. When writing the solution in interval
notation, if there are two different parts to the graph, a ∪ (union)
symbol is used between the two sets.
Example 4:
Graph and write interval notation for the disjunction:
x > 4 or x < -2
(-∞, -2] ∪ (4, ∞)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5:
Solve, graph, and write interval notation:
2x + 7 < -11 or -3x – 2 < 13
-7 -7
+2 +2
2x
< -18
-3x
< 15
2
2
-3
-3
x < -9
x > -5
Solve each inequality
Subtract or add
Divide
Dividing by a negative flips
the inequality sign
Notice when graphing, an open dot was used rather than the parentheses.
Unless indicated, either can be used.
(-∞, -9) ∪ (-5, ∞)
Example 6:
Solve, graph, and write interval notation:
x – 9 < -8 or 2x + 3 > -3
+9 +9
-3 -3
x
<1
2x
> -6
2
2
x > -3
Solve each inequality
Add or subtract
Divide
When the graphs are combined, the solution covers the entire number
line, so the interval notation is (-∞, ∞).
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
The second type of compound is an AND inequality. AND
inequalities require both statements to be true. If one is false, they
both are false. This compound inequality is called a conjunction.
The solution set of a conjunction is the intersection of the solution
sets of the individual inequalities.
When graphing these inequalities, follow a similar process as with
as above – first graph both inequalities above the number line, but
where the two inequalities overlap will be drawn onto the number
line for the final graph. When writing the solution in interval
notation, a ∩ (intersection) symbol could be used but is not
necessary and not commonly used.
Example 7:
Graph and write interval notation for the conjunction:
x > 2 and x < 5
(2, 5]
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 8:
Solve, graph, and write interval notation:
2x + 8 > 5x – 7 and 5x – 3 > 3x + 1
-5x
- 5x
-3x
-3x
-3x + 8 > - 7
2x – 3 > 1
-8
-8
+3 +3
-3x
< -15
2x
> 4
-3
-3
2
2
x < 5
x> 2
Solve each inequality
Move variable to one side
Subtract or add
Divide
Dividing by a negative flips
the inequality sign
(2, 5]
Example 9:
Solve, graph, and write interval notation:
x + 7 < 10 – 4 and 5x – 7 > 3
x+7<6
5x – 7 > 3
-7 -7
+7+7
x
< -1
5x
> 10
5
5
Solve each inequality
Combine like terms
Subtract or add
Divide
There is no overlap of the two inequalities, so there is no solution or Ø.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Conjunctions may also be expressed as a single sentence with the
variable (or expression containing the variable) between two
numbers. When solving these conjunctions, because there are
three parts (a tripartite), to stay balanced, the same thing must be
done to all three parts (rather than just both sides) to isolate the
variable in the middle. The graph then is simply the values
between the numbers with appropriate indication on the ends.
Example 10:
Graph and write interval notation for the conjunction:
-9 < x < 1
Notice when graphing, a solid dot was used rather than the bracket.
Unless indicated, either can be used.
[-9, 1]
Example 11:
Solve, graph, and write interval notation:
2 > -4x + 2 > -6
-2
- 2 -2
Subtract 2 from all three parts
0 > -4x > -8
Divide all three parts by -4
-4
-4
-4
0<
x
< 2
Dividing by a negative flips the
inequality signs
(0, 2]
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)