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2-8 Solving Absolute-Value Equations
and Inequalities
Objectives:
Solve compound inequalities.
Write and solve absolute-value equations and
inequalities.
A conjunction is a compound
statement that uses the word and.
Conjunction:
Set builder
notation:
x  3 AND x  2
x x  3
x  2
A conjunction is true iff all of its parts
are true.
A disjunction is a compound
statement that uses the word or.
Disjuncion: x  3 OR x  2
Set builder
x x  3
notation:

x  2
A disjunction is true if and only if
at least one of its parts is true
Example
Solve the compound
inequality. Then graph the
solution set.
6 y   24 OR y  5  3
–6 –5 –4 –3 –2 –1
0
1
2
3
Example
Solve the compound
inequality. Then graph the
solution set.
x  5  2 OR  2 x  10
–6 –5 –4 –3 –2 –1
0
1
2
3
Example
Solve the compound
inequality. Then graph the
solution set.
1
c  2 AND 2c  1  1
2
–6 –5 –4 –3 –2 –1
0
1
2
3
Helpful Hint
Think: Greator inequalities involving > or ≥
symbols are disjunctions.
Think: Less thand inequalities involving < or
≤ symbols are conjunctions.
Example
Solve the equation.
3  k  10
1. Set up two equations: -3 + k = 10 and -3 + k = -10
2. Solve both:
+3
+3
+3
+3
k = 13 and
k = -7
So there are two solutions to this equation: -7 and 13.
Example
x
 6  2
4
Solve the equation.
(Remember to isolate the
absolute value first before
you set up the two
equations.)
x
1. You can’t set up your two equations until you have just
by itself on one
4
side. So add 6 to both sides first.
x
 6  2
4
+6 +6
x
4
4
2. Now set up your two equations:
3. Solve both for x:
x
4
4
and
x
 4
4
x = 16
and
x = -16
Example
4q  2  10
Solve the inequality. Then
graph the solution.
You’ll solve it the same way as you solve an
equation, just with the additional graphing step
–6 –5 –4 –3 –2 –1
0
1
2
3
Example
Solve the inequality. Then
graph the solution.
ISOLATE 1ST!!!
0.5r  3  3
–6 –5 –4 –3 –2 –1
0
1
2
3
Example
Solve the compound
inequality. Then graph the
solution set.
ISOLATE 1ST!!!
1
 p2 3
2
–10
–5
0
5
10
15
20
25
Example
Solve the compound
inequality. Then graph the
solution set.
ISOLATE 1ST!!!
–10
2x  7
3
1
–5
0
5
10
15
20
25