Unit 1 Expressions, Equations and Inequalities

```Unit 1
Expressions,
Equations and
Inequalities
1.7 Linear Inequalities
and Absolute Value
Inequalities
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1
Objectives:
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Use interval notation.
Find intersections and unions of intervals.
Solve linear inequalities.
Recognize inequalities with no solution or all real
numbers as solutions.
Solve compound inequalities.
Solve absolute value inequalities.
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Solving an Inequality
Solving an inequality is the process of finding the set
of numbers that make the inequality a true statement.
These numbers are called the solutions of the inequality
and we say that they satisfy the inequality. The set of
all solutions is called the solution set of the inequality.
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Interval Notation
The open interval (a,b) represents the set of real
numbers between, but not including, a and b.
(a, b)   x a  x  b
The closed interval [a,b] represents the set of real
numbers between, and including, a and b.
[a, b]   x a  x  b
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Interval Notation (continued)
The infinite interval (a, ) represents the set of real
numbers that are greater than a.
(a, )   x x  a
The infinite interval ( , b] represents the set of real
numbers that are less than or equal to b.
(, b]   x x  b
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Parentheses and Brackets in Interval Notation
Parentheses indicate endpoints that are not included in
an interval. Square brackets indicate endpoints that are
included in an interval. Parentheses are always used
with  or  .
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Example: Using Interval Notation
Express the interval in set-builder notation and graph:
[1, 3.5]
x 1  x  3.5
Express the interval in set-builder notation and graph:
(, 1)
x x  1
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Finding Intersections and Unions of Two 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.
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Example: Finding Intersections and Unions of Intervals
Use graphs to find the set: [1,3]  (2,6)
Graph of [1,3]:
Graph of (2,6):
Numbers in both [1,3] and (2,6):
Thus,
[1,3]  (2,6)  (2,3]
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Solving Linear Inequalities in One Variable
A linear inequality in x can be written in one of the
following forms  a  0  :
ax  b  0
ax  b  0
ax  b  0
ax  b  0
In general, when we multiply or divide both sides of
an inequality by a negative number, the direction of
the inequality symbol is reversed.
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Example: Solving a Linear Inequality
Solve and graph the solution set on a number line:
2  3x  5
2  3x  5
3 x  3
3x 3

3 3
x  1
The solution set is
x x  1 .
The interval notation for this
solution set is [ 1, ) .
The number line graph is:
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Example: Solving Linear Inequalities
[Recognize inequalities with no solution or all real
numbers as solutions]
Solve the inequality: 3( x  1)  3 x  2
3( x  1)  3 x  2
3x  3  3x  2
32
The inequality is true for all values of x. The solution set
is the set of all real numbers.
In interval notation, the solution is (, )
In set-builder notation, the solution set is
x x is a real number
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Example: Solving a Compound Inequality
Solve and graph the solution set on a number line:
1  2 x  3  11
Our goal is to isolate x in the middle.
1  2 x  3  11
In interval notation,
2  2 x  8
the solution is [-1,4).
1  x  4
In set-builder notation, the solution set is
x 1  x  4
The number line graph looks like
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Solving an Absolute Value Inequality
If u is an algebraic expression and c is a positive
number,
1. The solutions of u  c are the numbers that satisfy
c  u  c
2. The solutions of u  c are the numbers that satisfy
u  c
uc
or
These rules are valid if  is replaced by  and
 is replaced by  .
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Example: Solving an Absolute Value Inequality
Solve and graph the solution set on a number line:
18  6  3x
We begin by expressing the inequality with the
absolute value expression on the left side:
6  3x  18
We rewrite the inequality without absolute value bars.
6  3x  18 means 6  3 x  18 or 6  3 x  18
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Example: Solving an Absolute Value Inequality
(continued)
We solve these inequalities separately:
6  3 x  18
6  3 x  18
3 x  24
3 x  12
3x 12
3x 24


3
3
3 3
x  4
x 8
The solution set is
x x  4 or x  8
The number line graph looks like
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