Algebra II :
Solving absolute value inequalities
.
Inequalities

less than
x  5
5

greater than
x5
5

less than or equal to
x  3

greater than or equal to
x 3
3
3
Conjunctions and Disjunctions
Conjunction (and)
3  x  3  3  x and x  3
3
3
Disjunction (or)
x  5 or x  5
3
3
5
5
Solving Absolute Value Inequalities
Method 1
Using understanding of conjunctions and disjunctions
1.
2.
Isolate the absolute value.
Expand into a conjunction or disjunction.
a)
b)
3.
4.
|ax + b| < c
|ax + b| > c
conjunction (AND)
disjunction (OR)
Solve. Remember to flip if multiplying by
a negative.
Graph.
x 3 2  7
1. Isolate the absolute value
5  x  3  5
2. Since it is a less than sign,
it is a conjunction (AND).
3. Focus on the middle
when solving.
x 3 5
subtract 2.
-2  x  8
add 3
-2
8
2 x  3  6
1. Isolate the absolute value.
2. Since it’s a greater than
or equal to sign , it’s a
disjunction (OR).
Notice
•The inside didn’t change
x3 3
divde by -2
x3 3
x  3  3
or
x3 3
or
x0
x  3  3
x  6
•Once is like the original problem
•The other is flip and make negative
3. Solve each separately.
-6
0
Solving Absolute Value Inequalities
Method 2
Using Critical Values and Interval Analysis





Change the inequality to equals and solve using the
directions for solving Absolute Value Equations.
Plot the answers (critical values) on a number line.
Choose the type of circle based on the original problem.
Test 1 number (not a critical value) to determine the
truth/falseness of 1 interval.
Except for special cases, intervals will alternate between
true and false. Shade where true.
Create your answer from your graph.
x 3 2  7
x 3 2  7
1. Change to = and solve
x 3  5
isolate
x35
x 8
split
solve
x  3  5
x  2
0
2. Plot the critical values
3. Test a number in the
original. Shade where true.
4. Create the answer from
the graph.
03 2 7
3 2  7
5  7 true
-2
<
x
<
8
2 x  3  6
2 x  3  6
1. Change to = and solve
x3 3
isolate
split
x33
x  3  3
x0
x  6
solve
1
2. Plot the critical values
3. Test a number in the
original. Shade where true. 2 1  3  6
8  6
4. Create the answer from
the graph.
x
true

-6
or
0

x