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Compound Inequalities
A compound inequality is a sentence with two
inequality statements joined either by the word “or” or
by the word “and.” “And” indicates that both
statements of the compound sentence are true at the
same time. It is the overlap or intersection of the
solution sets for the individual statements. “Or”
indicates that, as long as either statement is true, the
entire compound sentence is true. It is the
combination or union of the solution sets for the
individual statements.
3 x + 2 < 14 and 2 x – 5 > –11
2 x + 7 < –11 or –3 x – 2 < 13
or
Disjunction
1. Graph each inequality .
_l_____l______l_____l_____l______l______l_____l_____
-4
-3
-2
-1
0
1
2
3
4
Everything that is mentioned in the two inequalities is a
solution.
The set of all “x”s such that x is less than -3 or x is greater than 2.
and
Conjunction
Graph Each Inequality
l
l
-1
l
0
l
1
l
2
l
3
l l
4 5
l
6
l
And means intersection, the solution includes what the inequalities
have in common (the overlap)
l
l
l
l
l
-1
0
1
2
{a: 1 < a < 3}
l
3
l
4
l
5
l
6
l
The set of all “a”s such that 1 is less than
or equal to a which is less than or equal to 3.
Example 1: Solve 3x + 1< 4 or 2x – 5 > 7
1. Solve each inequality:
3x + 1 < 4
2x – 5 > 7
x<1
x>6
Graph the solution {x: x <1 or x > 6}
l l l l l l l
-6 -5 -4 -3 -2 -1 0
l
1
l
2
l
3
l
4
l
5
l
6
This compound inequality
can be interpreted as
1 < 2c – 7 < 7
1 < 2c – 7 and 2c – 7 < 7
c>4
l
l
l
l
l
l
l
2
3
4
5
6
7
8
c<7
Solve the compound inequality.
-2 < 2x + 6 < 12
Solve the inequalities
Graph the solutions
Determine the final solution
Express your solution in set notation
-2 < 2x + 6 < 12
-2 < 2x + 6
-6
-6
-8 < 2x
2 2
-4 < x
x > -4
l l l l
-4 -3 -2 -1
2X + 6 < 12
-6 -6
2X
< 6
2
2
x
<3
l
0
l
1
l l
2 3
l
4
2x + 3 < 1 or 3x – 5 > 4
2x + 3 < 1 or 3x – 5 > 4
2x + 3 < 1
- 3 -3
2x < -2
2
2
x < -1
3x – 5 > 4
+ 5 +5
3x
>9
3
3
x
>3
l l l
-4 -3 -2
l l l
-1 0 1
l
2
l
3
l
4