Compound Inequalities
Two types of compound inequalities we will study:
1. The Solution Set is contained between two numbers on the number line.
Ex:
All real numbers > 0 AND < 4
Symbol:
0 x4
Graph:
-----------------------------
Solution Set: {x: x = 0, 1, 2, 3}
a < x < b read as “x is between a and b.”
a  x  b read as “x is between a and b, inclusive.”
2. Solution Set is found less than one number or greater than another number on the
number line.
Ex:
All real numbers < -1 OR > 2
Symbol:
x  1 or x  2
Graph:
-----------------------------
Examples: Graph the following.
1) 2  x  8
2) 6  x  12
4)  2  x  3
5) 3  x  2
7) x  6 or x  0
8) x  2 or x  1
3)  6  x  4
6) x  5 or x  4
SOLVING COMPOUND INEQUALITIES:
When solving equations, we follow the rule, “What you do to one side, you must do to
the other.” The same applies to solving inequalities.
Ex:
 2  3x  8  10
+8
+8 +8
6  3x  18
3 3
3
2 x6
We must isolate x in the center so we can easily
graph the inequality.
-----------------------------
Q: Why perform the operation to both sides?
Always treat compound inequalities as two separate simple inequalities to make
solving simpler.
Substitute any number of your Solution Set into the original inequality to check the
graph.
 2  3x  8  10
 2  3(4)  8  10
 2  12  8  10
 2  4  10 ☺
Ex:
Ex:
3x  1  4
-1 -1
3x < 3
3 3
X<1
or
 2  2  x  1
+2 +2
+2
0 < -x < 3
-1
-1 -1
0 > x > -3
2x  5  7
+5 +5
2x > 12
2 2
x>6
-----------------------------
-----------------------------
REMEMBER: When multiplying or dividing by a negative, reverse the inequality sign.
Examples: Solve and Graph the following.
1)  6  x  4  4
2)  5  x  6  4
3)  4  3  x  2
4)  10  6  2x  8
5)  10  4x  18  30
6) 7  3x  4  19
7) 6  2x  20 or 8  x  0 8)  3x  7  8 or  2x  11  31
9)  8  1  3( x  2)  13
10) x is greater than 4 but at least 0.
11) x is between -9 and 2
12) x is less than 2 but is at least -7
Homework: Handout