* Use interval notation * Use properties of inequalities. * Solve inequalities in one variable algebraically. * Solve absolute value inequalities.

1. 2.7 Linear & Absolute Value
Inequalities
Chapter 2 Equations and Inequalities

2. Concepts and Objectives
⚫ Objectives for this section are:
⚫ Use interval notation
⚫ Use properties of inequalities.
⚫ Solve inequalities in one variable algebraically.
⚫ Solve absolute value inequalities.

3. Inequalities
⚫ An inequality states that one expression is greater than,
greater than or equal to, less than, or less than or equal
to another expression.
⚫ As with equations, a value of the variable for which the
inequality is true is a solution of the inequality; the set of
all solutions is the solution set of the inequality.
⚫ Inequalities are solved in the same manner equations
are solved with one difference—you must reverse the
direction of the symbol when multiplying or dividing by
a negative number.

4. Linear Inequalities
Example: Solve − +  −
2 7 5
x

5. Linear Inequalities
Example: Solve
The solution set is {x | x > 6}. Graphically, the solution is
− +  −
2 7 5
x
− +  −
2 7 5
x
7 7
2 7 5
x
− −
+ − −

−  −
2 12
x

− −
− −
2 12
2 2
x
 6
x

6. Intervals
⚫ The solution set for the inequality in the previous
example, {x | x < 6}, is an example of an interval.
⚫ We use a simplified notation, cleverly called interval
notation, to write intervals. With this notation, we
would write the interval in the example as (−, 6).
⚫ The symbol − does not represent an actual number; it
is used to show that the interval represents all real
numbers less than 6. This is an example of an open
interval, since the endpoint, 6, is not part of the interval.
A square bracket is used to show that a number is part of
the interval.

7. Intervals (cont.)
b
a
b
a
b
a b
a
b
a
b
a
Name of
Interval
Notation
Inequality
Description
Number Line Representation
finite, open
(a, b) a < x < b
finite, closed
[a, b] a x b
finite, half-
open
(a, b]
[a, b)
a < x b
a x < b
infinite, open
(a, )
(- , b)
a < x <
- < x < b
infinite,
closed
[a, )
(- , b]
a x <
- < x b
b
a
b
a
b
a
b
a
b
a
b
a

8. Compound Inequalities
⚫ Three-part or Compound Inequalities are solved by
working with all three expressions at the same time.
⚫ The middle expression is between the outer expressions.
⚫ Example: Solve  − 
1 6 8 4
x

9. Compound Inequalities (cont.)
⚫ Example: Solve
The solution set is the interval
 − 
1 6 8 4
x
8
1 8
8 8
6 4
x −
+  
+ +
 
9 6 12
x
6 6 6
9 6 12
x
 
 
3
2
2
x
 
 
 
3
,2
2

10. Absolute Value Inequalities
⚫ For absolute value inequalities, we make use of the
following two properties:
⚫ |a| < b if and only if –b < a < b.
⚫ |a| > b if and only if a < –b or a > b.
⚫ Example: Solve − + 
5 8 6 14
x

11. Absolute Value Inequalities (cont.)
⚫ Example: Solve
The solution set is
− + 
5 8 6 14
x
or
− 
5 8 8
x
−  −
5 8 8
x − 
5 8 8
x
−  −
8 13
x − 
8 3
x

13
8
x −

3
8
x
   
− −  
 
 
   
3 13
, ,
8 8

12. Special Cases
⚫ Since an absolute value expression is always
nonnegative:
⚫ Expressions such as |2 – 5x| > –4 are always true. Its
solution set includes all real numbers, that is, (–, ).
⚫ Expressions such as |4x – 7| < –3 are always false—
that is, it has no solution.
⚫ The absolute value of 0 is equal to 0, so you can solve
it as a regular equation.

13. Classwork
⚫ College Algebra 2e
⚫ 2.7: 6-22 (even); 2.6: 20-28 (even); 2.5: 32-42 (even)
⚫ 2.7 Classwork Check
⚫ Quiz 2.6