Download Students Matter. Success Counts.

Section 3.4
Linear Inequalities
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Objectives
o
o
o
o
Understand and use set-builder notation.
Understand and use interval notation.
Solve linear inequalities.
Solve compound inequalities.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Sets and Set-Builder Notation
Notes
Special Comments about Union and Intersection
The concepts of union and intersection are part of set
theory which is very useful in a variety of courses
including abstract algebra, probability, and statistics.
These concepts are also used in analyzing inequalities
and analyzing relationships among sets in general.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Sets and Set-Builder Notation
Notes (cont.)
The union (symbolized , as in A  B) of two (or more)
sets is the set of all elements that belong to either one
set or the other set or to both sets. The intersection
(symbolized , as in A  B) of two (or more) sets is the
set of all elements that belong to both sets. The word
or is used to indicate union and the word and is used to
indicate intersection. For example, if A = {1, 2, 3} and
B = {2, 3, 4}, then the numbers that belong to A or B is
the set A  B = {1, 2, 3, 4}. The set of numbers that
belong to A and B is the set A  B = {2, 3}.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Sets and Set-Builder Notation
Notes (cont.)
These relationships can be illustrated using the
following Venn diagram.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Sets and Set-Builder Notation
Notes (cont.)
Similarly, union and intersection notation can be used
for sets with inequalities.
For example,  x x  a or x  b can be written in the
form
x x  a  x x  b.
Also, x x  a and x  b can be written in the form
x x  a  x x  b or x a  x  b.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Intervals of Real Numbers
Type of
Interval
Open
Interval
Closed
Interval
Algebraic Interval
Notation Notation
a<x<b
(a, b)
a≤x≤b
[a, b]
Half-open a  x  b
Interval a  x  b
 a , b
 a , b
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Graph
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Intervals of Real Numbers
Type of
Interval
Algebraic Interval
Notation Notation
Open
Interval
x  a

x  b
 a, 
 , b
Half-open
Interval
x  a

x  b
 a, 
 , b
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Graph
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Intervals of Real Numbers
Notes
The symbol for infinity  (or ) is not a number. It is
used to indicate that the interval is to include all real
numbers from some point on (either in the positive
direction or the negative direction) without end.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 1: Graphing Intervals
a. Graph the open interval  3,  .
Solution
b. Graph the half-open interval 0  x  4.
Solution
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 1: Graphing Intervals (cont.)
c. Represent the following graph using algebraic
notation, and state what kind of interval it is.
Solution x ≥ 1 is a half-open interval.
d. Represent the following graph using interval
notation, and state what kind of interval it is.
Solution (3, 1) is an open interval.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 2: Sets of Real Numbers Illustrating
Union
Graph the set {x|x > 5 or x ≤ 4}. The word or implies
those values of x that satisfy at least one of the
inequalities.
Solution x > 5
x≤4
x > 5 or x ≤ 4
The solution graph shows the union  of the first two
graphs.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 3: Sets of Real Numbers Illustrating
Intersection
Graph the set {x|x ≤ 2 and x ≥ 0}. The word and implies
those values of x that satisfy both inequalities.
Solution
x≤2
x≥0
x ≤ 2 and x ≥ 0
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 3: Sets of Real Numbers Illustrating
Intersection (cont.)
The solution graph shows the intersection  of the first
two graphs. In other words, the third graph shows the
points in common between the first two graphs in this
example.
This set can also be indicated as {x|0 ≤ x ≤ 2}.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Solving Linear Inequalities
Linear Inequalities
Inequalities of the given form, where a, b, and c are
real numbers and a ≠ 0,
ax + b < c and ax + b ≤ c
ax + b > c and ax + b ≥ c
are called linear inequalities.
The inequalities c < ax + b < d and c ≤ ax + b ≤ d are
called compound linear inequalities. (This includes
c < ax + b ≤ d and c ≤ ax + b < d as well.)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Solving Linear Inequalities
Rules for Solving Linear Inequalities
1. Simplify each side of the inequality by removing any
grouping symbols and combining like terms.
2. Use the addition property of equality to add the
opposites of constants or variable expressions so
that variable expressions are on one side of the
inequality and constants are on the other.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Solving Linear Inequalities
Rules for Solving Linear Inequalities (cont.)
3. Use the multiplication property of equality to
multiply both sides by the reciprocal of the
coefficient of the variable (that is, divide both sides
by the coefficient) so that the new coefficient is 1. If
this coefficient is negative, reverse the sense of the
inequality.
4. A quick (and generally satisfactory) check is to select
any one number in your solution and substitute it
into the original inequality.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities
Solve the following linear inequalities and graph the
solution set. Write the solution set using interval
notation. Assume that x is a real number.
a. 6x + 5 ≤ 1
Solution
6 x  5  1
Write the inequality.
6 x  5  5  1  5
Add 5 to both sides.
6 x  6
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Simplify.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
6 x 6

6
6
x  1
x is in  , 1
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Divide both sides by 6.
Simplify.
Use interval notation. Note that the interval
(, 1] is a half-open interval.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
b. x  3  3x  4
Solution
x  3  3x  4
Write the inequality.
x  3  x  3x  4  x
Add x to both sides.
3  2x  4
3  4  2 x  4  4
Simplify.
Add 4 to both sides.
7  2x
Simplify.
7 2 x

2
2
Divide both sides by 2.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
7
x
2
or
7

x is in  ,  
2

HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Simplify.
7
x
2
Use interval notation. Note that the interval
7

  ,   is an open interval.
2

Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
c. 6  4 x  x  1
Solution 6  4 x  x  1
6  4x  x  x  1  x
6  5x  1
6  5x  6  1  6
5x  5
5x 5

5 5
x 1
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Write the inequality.
Add x to both sides.
Simplify.
Add 6 to both sides.
Simplify.
Divide both sides by 5. Note the
reversal of the inequality sign!
Simplify.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
x is in 1,  .
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Use interval notation. Note that the
interval [1, ) is a half-open interval.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
d. 2x  5  3x   7  x 
Solution
2 x  5  3x   7  x 
2 x  5  3x  7  x
2x  5  4 x  7
2x  5  2x  4 x  7  2x
5  2x  7
5  7  2x  7  7
12  2x
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Write the inequality.
Distribute the negative sign.
Combine like terms.
Add 2 x to both sides.
Simplify.
Add 7 to both sides.
Simplify.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 4: Solving Linear Inequalities (cont.)
12 2 x

2
2
6 x
x is in  6,  
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Divide both sides by 2.
Simplify.
Use interval notation. Note that the interval
(6, ) is an open interval.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
a. Solve the compound inequality 5 ≤ 4x  1 < 11 and
graph the solution set. Write the solution set using
interval notation. Assume that x is a real number.
Solution 5  4 x  1  11
Write the inequality.
5  1  4 x  1  1  11  1 Add 1 to each part.
4 
4

4
1 
4x
4x
4
x
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
 12
12

4
3
Simplify.
Divide each part by 4.
Simplify.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
(cont.)
The solution set is the half-open interval [1, 3).
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
(cont.)
b. Solve the compound inequality 5 ≤ 3 − 2x ≤ 13 and
graph the solution set. Write the solution set using
interval notation. Assume that x is a real number.
Solution 5  3  2x
 13
5  3  3  2x  3  13  3
8
 2x
 16
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Write the inequality.
Add 3 to each part.
Simplify.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
(cont.)
 or
8
2 x 16


2
2
2
Divide each part by 2. Note that
4  x
 8
Simplify.
8  x
 4 
the inequalities change sense.
The solution set is the closed interval [8, 4].
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
(cont.)
3x  5
 3 and
4
graph the solution set. Write the solution set using
interval notation. Assume that x is a real number.
c. Solve the compound inequality 0 
Solution
3x  5
0
3
4
3x  5
04 
 4  3 4
4
0  3x  5  12
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Write the inequality.
Multiply each part by 4.
Simplify each part.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 5: Solving Compound Inequalities
(cont.)
0  5  3x  5  5  12  5
5
3x
 17
5
3x
17


3
3
3
5
17

x

3
3
Add 5 to each part.
Simplify.
Divide each part by 3.
Simplify.
 5 17 
The solution set is the open interval  ,  .
3 3 
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6: Application with an Inequality
A math student has grades of 85, 98, 93, and 90 on four
examinations. If he must average 90 or better to
receive an A for the course, what scores can he receive
on the final exam and earn an A? (Assume that the
final exam counts the same as the other exams.)
Solution
Let x = score on final exam.
The average is found by adding the
scores and dividing by 5.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 6: Application with an Inequality (cont.)
85  98  93  90  x
 90
5
366  x
 90
Simplify the numerator.
5
 366  x 
5
 5  90
Multiply both sides by 5.

 5 
366  x  450
Simplify.
366  x  366  450  366 Add 366 to each side.
x  84
If the student scores 84 or more on the final exam, he
will average 90 or more and receive an A in math.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 7: Application with an Inequality
Ellen is going to buy 30 stamps, some 28-cent and
some 44-cent. If she has $9.68, what is the maximum
number of 44-cent stamps she can buy?
Solution
Let x = number of 44-cent stamps,
then 30 − x = number of 28-cent stamps.
Ellen cannot spend more than $9.68.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Example 7: Application with an Inequality (cont.)
0.44 x  0.28  30  x   9.68
0.44 x  8.40  0.28x  9.68
0.16x  8.40  9.68
0.16x  8.40  8.40  9.68  8.40
0.16x  1.28
0.16 x 1.28

0.16 0.16
x8
Ellen can buy at most eight 44-cent stamps if she buys a
total of 30 stamps.
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Practice Problems
Graph each set of real numbers on a real number line.
1.
3.
x x  3 and x  0
x x  2 or x  4
2.
x x  1.5
Solve each of the following inequalities and graph the
solution sets. Write each solution set in interval
notation. Assume that x is a real number.
4. 7  x  3
x
1
5.
1
2
3
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
6.  5  2 x  1  9
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
Practice Problem Answers
1.
2.
3.
4. (, 4)
4
5.  ,  
3 

6. [3, 4)
HAWKES LEARNING SYSTEMS
Students Matter. Success Counts.
Copyright © 2013 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.