Compound Inequalities
If we have two sets A and B, then we say that an element belongs to A
and
B if it belongs to both of them. For example, a student may belong
to the honor society if that student is a member of A and
B where
A = the collection of all registered students
B = the collection of all student has a GPA of at least
3.0.
Your instructor cannot belong to the honor society, since he is not a student.
Example
If
A =
{x | x > -2}
and
B =
{x | x <1}
then A and B is the in between part of the number line.
A and B = {x | -2 < x < 1}
Exercises
Write the following on a number line:
-
3x - 4 < 2 and 2x + 5 > 1
-
5x - 2 < 3 and 4x + 7 > 3
-
3x - 4 < 2 and 2x - 7 > 5
If we have two sets A and B, then we say that an element belongs to A
or
B is it belongs to either of them. For example, a person can get
a back seat pass to a concert if that person belongs to A or
B where
A =
set of people that know the band member personally.
B =
set of people that sit in the VIP seats.
We always have
(A and B) C
(A or B)
Example:
If
A = {x | x < -1)
B = {x | x > 2}
then A or B are the arrows left from
-1 and right from 2.
Exercises
Sketch the solution set on a number line.
-
3x + 2 > 4 or 5x + 7 < 2
-
x - 10 > 5 or 3x + 2 < 5
-
4x - 2 > 2 or 5x + 1 < 11
-
2x - 1 < 1 or 3x + 5 > 8
We also use intersection (I) for and and union (U)
for or.
Double Inequalities
We can write a compound inequality
as a double inequality as in the example
-3 <
2x + 5 < 7
This can be written as a compound inequality by writing
-3 <
2x + 5 and 2x + 5 < 7
-8 <
2x and 2x < 2
Subtracting 5
-4
< x and x < 1
Dividing by
2
Exercise
Write the double inequality as a compound inequality and then solve it.
-
-5 < 3x + 4 < 19
-
2y - 1 < y + 2 < 6y + 12
-
15 - t < t + 15 < 9t - 9
-
h + 1 < 2/3 h < h - 2
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