1.6 Compound and Absolute
Value Inequalities
Compound inequalities are just
more than one inequality at the
same time.
Sometimes, they are connected by AND.
Sometimes, they are connected by OR.
Solving an “AND” inequality
AND means “these two items must
both be true”. It is the intersection
of two inequalities.
Our strategy will be to solve the two
inequalities separately and then put the
solutions together.
Solving an “AND” inequality
Solve: 11 < 2x + 5 < 19
First, let’s find the two inequalities.
To find them, we just include:
1) The middle expression
2) An inequality symbol
3) Everything across the symbol from the
middle.
Solving an “AND” inequality
Solve: 11 < 2x + 5 < 19
11 < 2x + 5
2x + 5 < 19
6 < 2x
2x < 14
3<x
x<7
Now it’s time to put our
inequalities back together!
3<x<7
Graphing an “AND” inequality
Solve: 11 < 2x + 5 < 19
Solution:
3<x<7
3
3<x
x<7
3<x<7
4
5
6
7
Solving an “OR” inequality
OR means “at least one of the two
items must both be true”. It is the
union of two inequalities.
Our strategy will be to solve the two
inequalities separately and then put the
solutions together.
Solving an “OR” inequality
Solve: x - 2 > -3 or x + 4 < -3
Finding the two inequalities is a bit easier;
they are already listed separately.
Solving an “OR” inequality
Solve: x - 2 > -3 or x + 4 < -3
x - 2 > -3
x + 4 < -3
x > -1
x < -7
Now it’s time to put our
inequalities back together!
x > -1 or x < -7
Graphing an “OR” inequality
Solve:
x - 2 > -3 or x + 4 < -3
Solution: x > -1 or x < -7
-7
x > -1
x < -7
x > -1 or
x < -7
-6
-5
-4
-3
-2
-1
Graphing an Absolute Inequality
Graph:
|x| < 4
This means the distance from zero is < 4
-6 -4
-2
0
2
4
6
|x| < 4
Graph:
|x| > 4
This means the distance from zero is > 4
-6 -4
|x| > 4
-2
0
2
4
6
Solving Absolute Value Inequalities.
When we need to use two cases
to solve an absolute value
problem, treat the problem like an
OR inequality for graphing.