Today’s Topic: Compound Inequalities:
Conjunctions
This material accompanies the Make Math Matter podcast,
available for free download through USF on iTunes U at
http://itunes.usf.edu/. This document provides supplementary notes
on the full discussion from the podcast episode by the same name.
Note: We recommend that this be taught before
Disjunctions.
Introduce by discussing compound sentences. Thus, relating to prior knowledge.
Compound inequalities are two simple inequalities joined by “and” or “or”.
To solve the inequalities, you solve each one separately and graph the solutions on the
same number line.
Conjunctions are inequalities joined by “AND”
• True if both parts are true
• Could be written as two statements joined by “and” or can be written as one
combined statement with two inequality symbols
• Answer is the intersection (overlap) of the 2 inequalities Conjunction=Connect
• Solution must be possible
Graph and write as one statement. Stress that by convention, we put in order smallest to
largest and use < symbols.
1. x > 3 and x < 10
Ask yourself, “Is this possible?”
open circles on 3 and 10; overlap in between
3 < x and x < 10
3 < x < 10
2. x ! "4 and x # 2
Ask yourself, “Is this possible?”
closed circles on –4 and 2; overlap in between
!4 " x and x " 2
!4 " x " 2
3. x < 7 and x > 2
Ask yourself, “Is this possible?”
open circles on 2 and 7; overlap in between
x < 7 and x > 2
2 < x and x < 7
2< x<7
4. x > !4 and x < !6
Ask yourself, “Is this possible?”
no overlap
not possible to put together
x > !4 and x < !6
!6 > x and x < !4
!6 > x < !4
*You cannot have inequalities pointing on opposite directions
No solution to this problem – not possible
Solve and graph.
1. 3 x ! 1 > !28 and 2 x + 7 < 19
! (Null set, no solution)
3 x ! 1 > !28
2 x + 7 < 19
and
x > !9
x<6
Ask yourself, “Is this possible?”
!9 < x < 6
2. 7 ! 3 x " 2 ! 13
There are two ways to solve this problem.
I.
Rewrite as two statements joined by “and”
7 ! 3 x " 2 and 3 x " 2 ! 13
Ask yourself, “Is this possible?”
3 ! x and x ! 5
II.
3! x !5
Leave as is, and solve as one statement. Whatever you do to one part, you do
to the others.
7 ! 3 x " 2 ! 13
+2
+2 +2
9 ! 3x ! 15
3! x !5
Ask yourself, “Is this possible?”
Note: There are special cases, like x<3 and x<1 where overlap is at x<1.
Solution is x<1.
Review definition of compound inequality. Stress conjunctions, connect, “and”, must be
possible.