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Section 1.7 Linear Inequalities and Absolute Value Inequalities
*Interval Notation
Let a and b be real numbers such that a  b .
Interval Notation
Open Interval
(a, b)  {x | a  x  b}
Closed Interval
[a, b]  {x | a  x  b}
Infinite Interval
(a, )  {x | x  a}
Negative Infinite Interval
(, b]  {x | a  x  b}
Graph
a
b
x
a
b
x
a
x
b
x
Example 1) Complete the following table. (Table 1.5 on page 165)
Interval Notation Set-Builder Notation
Graph
( a, b)
{x | a  x  b}
a
b
x
(a, b]
{x | x  a}
a
(, b)
x
{x | x  b}
x
Example 2) Express each interval in set-builder notation and graph
a. [2, 5)
b. [1, 3.5]
c. (,  1)
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*Intersections and Unions of Intervals
A  B (A intersection B) is the set of elements common to both sets.
A  B (A union B) is the set of elements in set A or in set B or in both sets.
Finding Intersections and Unions of Intervals
1. Graph each interval on a number line.
2. a. To find the intersection, take the portion of the number line that the two graphs have
in common.
b. To find the union, take the portion of the number line representing the total
collection of numbers in the two graphs.
Example 3) Use graphs to find each set:
a. [1, 3]  (2, 6)
b. [1, 3]  (2, 6)
*Solving Linear Inequalities in One Variable
A linear inequality in x can be written in one of the following forms: ax  b  0 ,
ax  b  0 , ax  b  0 , ax  b  0 .
Solving a linear inequality is very similar to solving a linear equation, isolating a variable.
However, when we MULTIPLY or DIVIDE both sides of an inequality by a
NEGATIVE NUMBER, the direction of the inequality symbol is REVERSED.
Example 4) Solve and graph the solution set on a number line:
a. 2  3x  5
b. 3x  1  7 x  15
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*Inequalities with Unusual Solution Sets
Example 5) Solve each inequality:
a. 3( x  1)  3x  2
b. x  1  x  1
*Solving Compound Inequalities
Example 6) Solve and graph the solution set on a number line:
1  2x  3  11
*Solving Inequalities with Absolute Value
If X is an algebraic expression and c is a positive number,
1. The solution of X  c are the numbers that satisfy  c  X  c .
2. The solution of X  c are the numbers that satisfy X  c or X  c .
The rules are valid if  is replaced by  and > is replaced by  .
Example 7) Solve and graph the solution set on a number line
a. x  2  5
b.  3 5x  2  20  19
c. 18  6  3x
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