Page 1 (Section 1.1)
1.1 Linear and Rational Equations
In this section you will learn to:
• find restrictions on variable values
• solve linear equations in one variable
• solve rational equations with variables in the denominator
• recognize equations that are identities, conditional, or contradictions
• solve formulas for a specific value
Finding Variable Restrictions
Example 1: Find the restrictions on the variable a in each of the equations below.
(a)
5
2
3
5
2
+
=
−
a
a
a
(b)
a
a
a
a
5
4
3
3
2
−
=
−
−
Solving Linear Equations
(
0
=
+
bax , where a and b are real numbers and 0
≠
a )
Example 2:
Solve
xxx
−
=
−
6)6(3
Steps:
(if coefficients are integers)
1.
2.
3.
4.
Example 3:
Solve )53()22(5
−
+
=
+
−
xxxx
Page 2 (Section 1.1)
Example 4:
Solve: 1
3
2
5
3
+=
xx
(NOTE: This equation is
equivalent
to 1
3
2
5
3
+= xx .)
Steps:
(if coefficients are fractions)
1.
2.
3 – 6.
Solving Rational Equations
(equations having one or more “
rational
” expressions)
Example 5:
Solve 4
3
105
+=
x
x
Steps:
(if variable is in denominator)
1. Exclude any values that cause a zero
denominator.
2. Find the LCD and multiply both sides
by the LCD.
3. Simplify and solve the equation.
4. Check your solution in the
original
equation.
Example 6:
Find all values of x for which
21
yy =
given
16
22
2
1
−
=
x
y and
4
1
4
1
2
−
+
+
=
x
x
y .
Page 3 (Section 1.1)
Example 7:
Solve for n:
8
2
6
2
5
4
1
2
−
−
=
+
−
−
n
n
n
n
Example 8:
Solve:
3
2
1
1
3
2
2
−
+
−=
+
+
x
x
x
x
Types of Equations
Identity Conditional Contradiction/Inconsistent
Page 4 (Section 1.1)
Solving Formulas for a Specific Value
Steps:
Example 9:
Solve BhV
3
1
= for B. 1. If fractions are involved, multiply by LCD.
2. Move terms with desired variable to one
side. Move all other terms to other side.
3. If two or more terms contain the desired
variable, FACTOR out the variable.
4. Divide both sides by the “non variable”
factor.
Example 10:
Solve S = P + Prt for t.
Example 11:
Solve S = P + Prt for P.
Example 12:
Solve
c
ba
x
−
= for b.
Example 13:
Solve
z
w
yx
2
+= for z.
Example 14:
Solve
B
A
BA
C
−
+
= for A.
Example 15:
Solve 4
321
=−+
zyx
for x.
Page 5 (Section 1.1)
1.1 Homework Problems
1. Find restrictions on the variable x in each equation below:
(a)
4
52
2
−
=
x
x
(b)
43
1
)4(
3
2
−+
=
+
−
xx
xx
x
(c)
x
x
x
x
27
3
3
3
1
32
−
+
=
For Problems 2–8, solve the equations and classify each as an identity, contradiction or conditional.
2. 383)15(42
+
=
+
−
xxx 3.
10
1
5
2
1
2
=−−
xx
4.
2
13
4
1
3
13
=
−
−
+
xx
5. xxx 3)3(111014
+
+
=
+
6.
x
x
x
x
2
206
2
8
2
+
=+
+
7. )3(7217
−
=
−
xx
8.
3
2
1
1
3
2
2
−
+
−=
+
+
x
x
x
x
9. Solve rC
π
2
=
for r. 10. Solve cba 32
−
=
for c.
11. Solve 3=−
b
y
x
a
for x. 12. Solve
B
A
BA
C
−
+
= for B.
13. Solve
D
DAC
BA
2
+
=+ for D. 14. Solve:
c
b
a
321
=− for b.
15. Solve )32(
9
5
−= FC for F.
1.1 Homework Answers:
1.(a) 2,0
±
≠
x (b) 1,0,4
−
≠
x (c) 3,0
±
≠
x 2. {-2}; conditional
3. {2}; conditional 4.
15
77
; conditional 5.
φ
; contradiction 6.
7
4
; conditional
7. all real numbers; identity 8. {2}; conditional 9.
π
2
C
r =
10.
3
2
ab
c
−
=
11.
yb
ab
x
+
=
3
12.
1
+
−
=
C
AAC
B
13.
2
−+
=
B
A
AC
D
14.
a
c
ac
b
3
2
−
=
15. 32
5
9
+= CF
Page 1 (Section 1.2)
1.2 Applications of Linear Equations
In this section you will learn to:
• use linear equations to solve word problems
Example 1: The number of cats in the U. S.
exceeds the number of dogs by 7.5 million.
The number of cats and dogs combined is
114.7 million. Determine the number of dogs
and cats in the U. S.
Example 2: After a 30% discount, a cell phone
sells for $112. Find the original price of the cell
phone before the discount was applied to the
purchase.
Example 3: Including a 6% sales tax, an item
costs $91.69. Find the cost of the item before
the sales tax was added.
Example 4: You are choosing between two car
rental agencies. Avis charge $40/day plus $.10/mile
to rent a car. Hertz charges $50/day plus $.08/mile.
You plan to rent the car for three days. After how
many miles of driving will the total cost for each
agency be the same?
Steps/Tips for Solving Word Problems:
1. Read the problem carefully. Underline key words and phrases. Let x (or any variable)
represent one of the unknown quantities. Draw a picture or diagram if possible.
2. If necessary write expressions for other unknowns in terms of x.
3. Write a verbal model of the problem and then replace words with numbers, variables
and/or symbols.
4. Solve the equation. Answer the question in the problem. Label answers!
5. Check your answer(s) in the original word problem (not in your equation).
Page 2 (Section 1.2)
Example 5: The perimeter of a triangular lawn is
162 meters. The length of the first side is twice the
length of the second side. The length of the third side
is 6 meters shorter than three times the length of the
second side. Find the dimensions of the triangle.
Example 6: (Simple Interest Problem) Tricia received
an inheritance of $5500. She invested part of it at 8%
simple interest and the remainder at 12% simple interest.
At the end of the year she had earned $540. How much
did Tricia invest at each interest amount?
Example 7: How many liters of a 9% solution of salt
should be added to a 16% solution in order to obtain
350 liters of a 12% solution?
Example 8: A student scores 82%, 86% and 78% on her
first three exams. What score is needed on the fourth exam
for the student to have an average of 85% for all four exams?
Example 9: A student scores 85%, 72%, 96%, and 98% for
his first four chapter exams. If the fifth exam, the final exam,
counts twice as much as each of the chapter exams, is it possible
for the student to get a high enough final exam score to get a
90% average for the course?
Page 3 (Section 1.2)
Example 10: A rectangular swimming pool measures
18 feet by 30 feet and is surrounded by a path of uniform
width around all four sides. The perimeter of the rectangle
formed by the pool and the surrounding path is 132 feet.
Determine the width of the path.
Example 11: Sam can plow a parking lot in 45 minutes.
Eric can plow the same parking lot in 30 minutes. If Sam
and Eric work together, how long will it take them to clear
the lot?
Example 12: A plane leaving Lansing International Airport
travels due east at a rate of 500 mph. A second plane takes
off 15 minutes later traveling in the same direction at 650 mph.
How long will it take for the second plane to overtake the first?
Page 4 (Section 1.2)
1.2 Homework Problems
1. When a number is decreased by 30% of itself, the result is 56. What is the number?
2.
4
1
5 % of what number is 12.6? 3. 25 is what % of 80? 4. Find
.
25% of $240.
5. After a 20% price reduction, a cell phone sold for $77. Find the original price of the phone.
6. Find two consecutive even integers such that the sum of twice the smaller integer plus the larger is 344.
7. The length of a rectangular garden plot is 6 feet less than triple the width. If the perimeter of the field is
340 feet, what are its dimensions?
8. Trinity College currently has an enrollment of 13,300 students with a projected enrollment increase of
1000 students a year. Brown college now has 26,800 students with a projected enrollment decline of
500 students per year. Based on these projections, when will the colleges have the same enrollment?
9. Sam invested $16,000 in two different stocks. The first stock showed a gain of 12% annual interest
while the second stock suffered a 5% loss. If the total annual income from both investments was $1240,
how much was invested at each rate?
10. How many liters of a 7% acid solution should be added to 30 liters of a 15% solution in order to obtain a
10% solution?
11. How many liters of skim milk (0% fat) must be added to 3 liters of milk containing 3.5% butterfat in
order to dilute the milk to 2% butterfat?
12. Amy scored 78%, 64%, 98%, and 88% on her first four exams. What score does she need on her fifth
exam in order to have an 85% average for all five exams?
13. Scott showed improvement on his five math exams throughout the semester improving by 3% on each
successive exam. If the fifth exam (final exam) counted twice as much as the first four exams and his
average was 79% for all five exams, what score did he receive on his first exam?
14. The length of a rectangular tennis court is 6 feet longer than twice the width. If the perimeter of the
court is 228 feet, find the dimensions of the court.
15. During a road trip, Tony drove one-third the distance that Lana drove. Mark drove 24 more miles than
Lana. The total distance they drove on the trip was 346 miles. How many miles did each person drive?
16. A garden hose can fill a swimming pool in 5 days. A larger hose can fill the pool in 3 days. How long
will it take to fill the pool using both hoses?
17. The Smith family drove to their vacation home in Michigan in 5 hours. The trip home took only 3
hours since they averaged 26 mph more due to light traffic. How fast did they drive each way?
1.2 Homework Answers:
1. 80 2. 240 3. 31.25% 4. .6 5. $96.25 6. 114 and 116 7. 44 ft by 126 ft
8. 9 years 9. $12,000 @ 12%; $4000 @ 5% 10. 50 liters 11. 2.25 liters 12. 97% 13. 72% 14. 36 ft
78 ft 15. Tony: 46 miles; Lana: 138 miles; Mark:162 miles 16. 1.875 days (45 hrs) 17. 39 mph; 65 mph
Page 1 (Section 1.3)
1.3 Quadratic Equations
In this section you will learn to:
• solve quadratics equations by
1. factoring
2. square root property
3. quadratic formula
4. completing the square
5. graphing (used mainly for checking – not considered an algebraic solution)
• use the discriminant to find the number and type of solutions (roots, x-intercepts, zeros)
A
quadratic equation
in x is an equation that can be written in the
general form
0
2
=++ cbxax
,
where a, b, and c are real numbers, with
0
≠
a . A quadratic equation in x is also called a
second-degree polynomial equation.
The Zero
-
Product Principle:
If the product of two algebraic expressions is zero, then at least one of
the factors equal to zero.
If AB = 0, then A = 0 or B = 0
.
Solving Quadratic Equations by Factoring:
Steps:
Example 1:
xx 129
2
=
Example 2:
103
2
+= xx
1. Rearrange equation so
that one side is 0.
2. Factor. (Use sum/product
idea when a = 1. If
1
≠
a , use grouping*.)
3. Set each factor equal to 0.
4. Solve each equation.
5. Check in original
equation or by graphing
(observe x-intercepts).
*Refer to Page 17 of Appendix A at the back of your textbook for steps using “Grouping Method”.
Page 2 (Section 1.3)
Example 3:
3134
2
−=− xx (Use “guess & check” or “grouping method”.)
Solving Quadratic Equations by the Square Root Method:
Square Root Property:
If a > 0 then ax =
2
has two real roots: ax = or ax −=
Reminder: Any time you choose to take a square root when solving an equation, you must include
±
.
(Example: If 4
2
=x , then 2
±
=
x .)
Example 4:
4713
2
=−x
Example 5:
5)38(
2
=−x
Solving Quadratic Equations Using the Quadratic Formula:
Quadratic Formula:
If
0
2
=++ cbxax
,
where 0
≠
a , then
a
acbb
x
2
4
2
−±−
=
.
Example 6:
Solve and simplify: 153
2
−= xx
Page 3 (Section 1.3)
Example 7:
Solve and simplify: 13164
2
=+ xx
Example 8:
Solve for t and simplify:
2
1632 tth −=
In the quadratic formula,
a
acbb
x
2
4
2
−±−
=
, the value of
acb 4
2
−
is called the
discriminant.
Beware:
The discriminant is
NOT
acb 4
2
−
!!!
04
2
>− acb
04
2
<− acb
04
2
=− acb
Example 9:
Determine the number and type of solutions for the equations below. (Do not solve.)
(a) 2132
2
=− xx (b) xx 704925
2
=+
Page 4 (Section 1.3)
Solving a Quadratic Equation by Completing the Square:
Recall:
Perfect Square Trinomials 96)3(
22
++=+ xxx
22
)6(3612 +=++ xxx
22
)(___8 =+− xx
22
)(4_____ =++x
22
)(100_____ =+−x
Example 10:
0352
2
=−+ xx
Steps:
1. Divide each term by the leading
coefficient a.
2. Move the constant term to the
right side.
3. “Form” a perfect square trinomial
by adding
2
2
1
b to both sides.
4. Factor the left side (perfect square
trinomial).
Add the terms on the right side.
5. Finish solving using the Square
Root Method.
Page 5 (Section 1.3)
1.3 Homework Problems
Solve Problems 1-6 by factoring:
1. xx 53
2
=
2. xx 215
2
=−
3. 2510
2
+=−
xx
4. 3015)12(
=
−
−
aa 5. 3042
2
=−
xx 6. 673
2
+=
mm
Solve Problems 7-9 by using the quadratic formula:
7. xx 815
2
=+
8. 0184
2
=+−
xx 9. 724
2
+=
xx
Solve Problems 10-12 using the square root method:
10.
49
25
2
=x 11. 35)112(
2
=++x 12. 8)43(
2
=−x
13. Solve for t: 416
2
−= th 14. Solve for x: 1
2
2
2
2
=−
b
y
a
x
For Problems 15-17, determine the number and type of solutions by examining the discriminant.
15. xx 2213
2
−=− 16. 049202
2
=+− xx 17. xx 42499
2
=+
Solve each of the following quadratic equations by completing the square:
18. 0158
2
=++ xx 19. 103
2
=− xx 20. xx 874
2
−=
1.3 Homework Answers:
1.
3
5
0
2. {-3, 5} 3. {-5} 4. {-3, 15} 5. {-3, 5} 6.
− 3,
3
2
7. {3, 5} 8.
±
2
32
9.
±
4
291
10.
±
7
5
11.
φ
12.
±
3
224
13.
+
±
4
4h
14.
+
±
b
yba
22
15. 0 real roots 16. 2 real roots 17. 1 real root 18. {-5, -3} 19. {-2, 5}
20.
±−
2
112
Page 1 (Section 1.4)
1.4 Application of Quadratic Equations
In this section you will learn to:
• solve rational equations by changing to quadratic form
• use quadratic equations to solve word problems
Solving Rational Equations: Steps:
Example 1:
x
x
24
2 =− 1. Multiply both sides by the LCD.
2. Simplify both sides.
3. Change to quadratic form:
.0
2
=++ cbxax
4. Solve using a quadratic technique.
5. Check answer in original equation.
(Be aware of restrictions!)
Example 2:
4
5
4
1
1
1
=
−
+
−
x
x
Example 3:
The length of a rectangle exceeds its width
by 3 feet. If its area is 54 square feet, find its dimensions.
Page 2 (Section 1.4)
Example 4:
The MSU football stadium currently has the
4
th
largest HD video screen of any college stadium. The
rectangular screen’s length is 72 feet more than its height.
If the video screen has an area of 5760 square feet, find
the dimensions of the screen. (MSU math fact: The area
of the video screen is about 600 ft
2
larger than Breslin’s
basketball floor.)
Example 5:
When the sum of 8 and twice a positive
number is subtracted from the square of the number,
the result is 0. Find the number.
Example 6:
Find at least two quadratic equations
whose solution set is
− 5,
3
2
.
Example 7:
The height of an object thrown upward
from the roof of a building 200 feet tall, with an initial
velocity of 100 feet/second, is given by the equation
20010016
2
++−=
tth
, where
h
represents the height
of the object after
t
seconds. How long will it take the
object to hit the ground?
(Round answer to nearest hundredth.)
Page 3 (Section 1.4)
Example 8:
John drove his moped from Lansing to Detroit,
a distance of 120 km. He drove 10 km per hour faster on the
return trip, cutting one hour off of his time. How fast did he
drive each way?
Example 9:
In a round-robin tournament, each team is
paired with every team once. The formula below models
the number of games,
N
, that must be played in a tournament
with
x
teams. If 55 games were played in a round-robin
tournament, how many teams were entered?
2
2
xx
N
−
=
Example 10:
When tickets for a rock concert cost $15,
the average attendance was 1200 people. Projections showed
that for each 50¢ decrease in ticket prices, 40 more people
would attend. How many attended the concert if the total
revenue was $17,280?
Page 4 (Section 1.4)
1.4 Homework Problems
1. Solve:
2
1
3
11
+
−=
x
x
2. Solve:
1
2
1
1
1
+
−=
−
x
x
3. The base of a triangle exceeds its height by 17 inches. If its area is 55 square inches, find the base
and height of the triangle.
4. A regulation tennis court for a doubles match is laid out so that its length is 6 feet more than two
times its width. The area of the doubles court is 2808 square feet. Find the length and width of a
doubles court.
5. If 120 games were played in a round-robin tournament, how many teams were entered? (Refer to
Example 9 in class notes for formula.)
6. A quadratic equation has two roots: ¾ and -5. (a) Find a quadratic equation where the coefficient of
the
x
2
term is 1. (b) Find a second equation that has only integers as coefficients.
7. The height of a toy rocket launched from the ground with an initial velocity of 128 feet/second, is
given by the equation
tth
12816
2
+−= , where
h
represents the height of the rocket after
t
seconds.
How long will it take the rocket to hit the ground? (Round answer to nearest hundredth.)
8. The height of an object thrown upward from the roof of a building 200 feet tall, with an initial
velocity of 100 feet/second, is given by the equation 20010016
2
++−=
tth
, where
h
represents the
height of the object after
t
seconds. At what time(s) will the object be 300 feet above the ground?
(Round answer to nearest hundredth.)
9. Jack drove 600 miles to a convention in Washington D. C. On the return trip he was able to increase
his speed by 10 mph and save 3 hours of driving time. (a) Find his rate for each direction. (b) Find
his time for each direction.
10. When tickets for a rock concert cost $12, the average attendance was 500 people. Projections showed
that for each $1 increase in ticket prices, 50 less people would attend. At what ticket price would the
receipts be $5600.
1.4 Homework Answers:
1.
{
}
102 ± 2. {0, 3} 3. base: 22 in; height: 5 in 4. 78 ft by 36 ft
5. 16 teams 6. (a) 0
4
15
4
17
2
=−+
xx
; (b) 015174
2
=−+
xx
(answers vary) 7. 8 seconds
8. 1.25 and 5 seconds 9. (a) 40 mph; 50 mph (b) 15 hours; 12 hours 10. $14 ($2 increase)
Page 1 (Section 1.5)
1.5 Complex Numbers
In this section you will learn to:
• add, subtract, multiply and divide complex numbers
• simplify complex numbers
• find powers of i
• solve quadratic equations with complex roots
The imaginary unit i is defined as
1−=i
, where
1
2
−=i .
The set of all numbers in the form
a + bi
with real numbers a and b and i, the imaginary unit, is called
the set of
complex numbers
. (
Standard form
for a complex number is
a + bi
.)
THE COMPLEX NUMBER SYSTEM
Real
Numbers
Imaginary
Numbers
Rational Numbers
Irrational Numbers
Example 1:
Simplify each imaginary number below:
9−
= _____
36−
= _____
8−
= _____
27−
= ______
25
12−
= _____
Equality of Complex Numbers:
dicbia
+
=
+
if and only if
c
a
=
and
.db
=
Addition of Complex Numbers:
idbcadicbia )()()()(
+
+
+
=
+
+
+
Subtraction of Complex Numbers:
idbcadicbia )()()()(
+
−
+
=
+
−
+
Multiplication of Complex Numbers:
ibcadbdacdicbia )()())((
+
+
−
=
+
+
NOTE
: Add, subtract, or multiply complex numbers as if they were binomials.
Page 2 (Section 1.5)
Example 2:
Add: (2 – 3i) + (-4 + 5i)
Example 3:
Subtract: (2 – 3i) – (-4 + 5i)
Example 4:
Multiply: (2 – 3i)(-4 + 5i)
Recall:
(a + b) and (a – b) are c
onjugates.
Therefore, (a + b)(a – b) = a
2
– b
2
.
bia
+
and bia
−
are called
complex conjugates
therefore:
=
−
+
))(( biabia
Example 5 (Division):
Simplify
i
i
5
4
32
+−
−
(Hint: Multiply numerator and denominator by
conjugate
of the denominator.)
Example 6 (Division):
Find the reciprocal of
4 + 3i.
Page 3 (Section 1.5)
Example 7:
Simplify each of the following powers of i.
3
i = _____
4
i = _____
13
i = _____
102
i = _____
1−
i = _____
6−
i = _____
Recall:
i=−1
. Therefore
bibbb =−=−=− 1)1(
.
Example 8:
Simplify and write each answer in
standard form.
(a) 1827 −+− (b) 94 −⋅−
(c)
2
)52( −+− (d) )31)(31( −−−−+−
(e)
12
1212 −+−
Example 8:
Solve: 084
2
=++
aa
Example 9:
Solve: 55
2
−=+
xx
Page 4 (Section 1.5)
1.5 Homework Problems
1. Simplify each expression: (a) 169− (b) 200− (c) 108− (d)
49
50
−
(e)
4
27
6 −
(f)
5
i
(g)
26
i
(h)
7
−
i
2. Perform the indicated operation for each problem below. Simplify each problem and write the
answer in standard form.
(a) )48()73(
ii
−
−
+
−
(b) )59()226(
ii
+
−
−
−
(c) )58)(43(
ii
+
−
−
(d)
2
)54(
i
− (e)
i
5
(f)
i
i
+
−
1
(g)
i
i
+
−
3
2
(h)
3
)2(
i
+
(i) 8200 −+− (j) 2516 −⋅− (k) 273 −⋅−
(l)
2
)52(
i
−− (m)
2
)41(
1
−−
(n)
9
2718 −−−
3. Simplify:
i
i
2
1
5
)32(3
2
+
−+ 4. Simplify: )32)(32(
ixix
−
−
+
−
5. Find the reciprocal of 7 + 3
i
.
6. Solve each of the following:
(a) 172
2
−=
xx
(b)
xx
652
2
=+ (c) 08249
2
=+−
xx
(d) 4911
2
−=+
xx
1.5 Homework Answers:
1. (a) 13
i
(b)
210i
(c) 36
i
(d) 2
7
5
i
(e) 39
i
(f)
i
(g) -1
(h)
i
2. (a) -5 – 11
i
(b) 15 – 27
i
(c) -4 + 47
i
(d) -9 – 40
i
(e) -5
i
(f)
i
2
1
2
1
−−
(g)
i
2
1
2
1
− (h) 2 + 11
i
(i)
212i
(j) -20 (k) -9 (l) 21 + 20
i
(m)
i
25
4
25
3
+− (n) i
3
3
2 −−
3. -16 + 38i 4. 134
2
+− xx 5. i
58
3
58
7
− 6. (a) i41
±
; (b) i
2
1
2
3
± ; (c)
3
224 ±
or
3
22
3
4
± ;
(d) i
2
35
2
11
±−
Page 1 (Section 1.6)
1.6 Polynomial & Radical Equations
In this section you will learn to:
• solve polynomial equations using factoring
• solve radical equations
• solve equations with rational exponents
• solve equations in quadratic form
A polynomial equation in x:
)()( xqxp
=
, where p(x) and q(x) are polynomials.
The
general form of a polynomial is
p(x) = 0
The
degree of a polynomial
is the highest degree of any term in the polynomial.
Solving Polynomials by Factoring:
Example 1:
Solve:
24
124 xx =
Example 2:
Solve:
23
18489 yyy +=+
Example 3:
Solve: 062
23
=−− xxx
Solving Radical Equations:
Example 4:
Solve: 111 =+− xx
Steps:
1. Isolate radical(s).
2. Square both sides.
3. Expand.
4. Solve for x.
5.
Check answer!
Page 2 (Section 1.6)
Example 5:
Solve: 235 =−−+ xx (Hint: Move one of the radicals to the other side and repeat
Example 4 Steps 1-3 after expanding.)
Solving Equations in Quadratic Form (using substitution):
Example 6:
Solve: 0208 =−− xx
Example 7:
Solve: 9134
24
−= xx
Example 8:
Solve:
03116
5
1
5
2
=++ xx
Example 9:
Solve: 087
12
=−−
−−
xx
Page 3 (Section 1.6)
Example 10:
Try this alternative shortcut to substitution using this example for the following problems:
0103
2
=−− xx 250)2)(5(
−
=
=
⇒
=
+
−
⇒
xorxxx
0103
24
=−− xx
0103 =−− xx
0103
3
1
3
2
=−− xx
0103
12
=−−
−−
xx
010)2(3)2(
2
=−+−+ xx
Solving Equations with Rational Exponents:
Recall:
<−
≥
==
0
0
2
xifx
xifx
xx , and
2
1
xx =
,
3
1
3
xx =
,
3
2
3 2
xx =
, . . . . . .
m
n
m n
xx =
Example 11:
Solve: 0248
3
5
=−x
Example 12:
Solve: 4)5(
3
2
=+x
Page 4 (Section 1.6)
1.6 Homework Problems
Solve each of the equations below using any appropriate method:
1. 0255
35
=− xx 2.
23
56 xxx =+ 3.
3
2
12
−=− x
x
x
4. 012
24
=+− xx 5. x
x
xx
=+
−
− 3
1
2
2
3
6. xx =+− 132
7. 352 −=− xx 8. 065
3
1
3
2
=−+ xx 9. 015 =−++ xx
10. 455 =++− xx 11. 1132 +−=+ xx 12.
24
502 xx =
13.
23
1892 xxx =−+ 14. 0243
4
3
=−x 15. 4)7(
3
2
=−x
16. 623
23
+=+ xxx 17.
6
117
1
2
+−
=+
x
x
Solve each of the following equations using substitution:
18. 20
12
=−
−−
xx 19. 352 −=− xx 20. 087
2
3
3
=−− xx
1.6 Homework Answers:
1.
{
}
5,0 ± 2. {0, 2, 3} 3. {-2, 4} 4.
{
}
1±
5. {2, -3} 6. {2}
7.
1,
4
9
8. {-216, 1} 9.
φ
10.
{
}
4±
11. {-1} 12. {-5, 0, 5} 13.
− 3,
2
1
,3
14. {16}
15. {-1, 15} 16.
{
}
2,3 ±− 17.
−
2
1
,
3
5
18.
−
5
1
,
4
1
19.
1,
4
9
20. {4}
Page 1 (Section 1.7)
1.7 Inequalities
In this section you will learn to:
• use interval notation
• understand properties of inequality
• solve linear (and compound) inequalities
• solve polynomial inequalities
• solve rational inequalities
Interval Notation:
Inequality Graph Interval Notation
Set Builder Notation
4
<
x
3
−
≥
x
23
≤
<
−
x
3
−
≤
x
or
5
>
x
all real #’s
Intersection Union
Page 2 (Section 1.7)
Properties of Inequalities
+/- Property of Inequality Multiplication/Division Property of Inequality
If
ba
<
, then
cbca
+
<
+
cbca
−
<
−
(Adding or subtracting does not
affect the > or < sign.)
c > 0 (c is positive) c < 0 (c is negative)
If
ba
<
, then
bcac
<
c
b
c
a
<
(Multiplying or dividing by a
positive number does not affect
the > or < sign.)
If
ba
<
, then
bcac
>
c
b
c
a
>
(When multiplying or dividing by
a negative number, reverse the >
or < sign.)
Example 1: Solve and graph the inequality below. Example 2: Solve and graph the inequality below.
Write the answer using interval notation. Write the answer using interval notation.
532
≤
−
x
)5(13)197(5
−
>
+
−
xx
Example 3:
Solve the compound (“and”) inequality
51
3
2
3
≤+<− x
. Write answer using interval notation.
(a) Solve by isolating the variable
x
. (b) Solve by writing each inequality separately.
Example 4:
Avis charges $40/day plus $.10/mile to rent a car. Hertz charges $50/day plus $.08/mile.
When is Avis a better deal if you are renting a car for three days?
Page 3 (Section 1.7)
A
polynomial inequality
is any inequality of the form:
0)(
<
xf
(graph is below the
x
-axis)
0)(
≤
xf
(graph is on or below the
x
-axis)
0)(
>
xf
(graph is above the
x
-axis)
0)(
≥
xf
(graph is on or above the
x
-axis)
where
f
is a
polynomial
function
.
Example 5:
Solve
xx
56
2
>−
. Write the solution using interval notation.
Steps for Solving Polynomial Inequalities:
1. Express as
f
(
x
) > 0 or
f
(
x
) < 0.
(Get 0 on right side.)
2. Set
f
(
x
) = 0 and solve for
x
to get
Boundary Points.
3. Plot the boundary points on a number
line to obtain
Intervals.
4.
Test Values
within each interval and
evaluate )(
xf
for each value.
If )(
xf
> 0, then )(
xf
is + for interval.
If )(
xf
< 0, then )(
xf
is – for interval.
5. Write the solution using interval notation.
Check the solution on your calculator.
Example 6:
Solve
xxx
99
23
≤−+
. Write the solution using interval notation.
Page 4 (Section 1.7)
Example 7:
Solve 842
23
+≥+
xxx
. Write the solution using interval notation.
A
rational inequality
is any inequality of the form:
0)(
<
xf
(graph is below the
x
-axis)
0)(
≤
xf
(graph is on or below the
x
-axis)
0)(
>
xf
(graph is above the
x
-axis)
0)(
≥
xf
(graph is on or above the
x
-axis)
where
f
is a
rational function
. (
)(
)(
)(
xq
xp
xf
=
, where
p
and
q
are polynomials and 0)(
≠
xq
)
Example 8:
Solve 0
5
2
≥
+
−
x
x
. Write the solution using interval notation.
Steps for Solving Rational Inequalities:
1. Express as )(
xf
> 0 or )(
xf
< 0.
(Get 0 on right side.)
*2. Find values that make the numerator &
the denominator = 0. These are the
Boundary Points. (Note Restrictions!)
3. Plot the boundary points on a number
line to obtain
Intervals.
4.
Test Value
within each interval and
evaluate )(
xf
for each value.
If )(
xf
> 0, then )(
xf
is + for interval.
If )(
xf
< 0, then )(
xf
is – for interval.
5. Write the solution using interval notation.
Check the solution on your calculator.
Page 5 (Section 1.7)
Example 9:
Solve 2
2
≥
+
x
x
. Write the solution using interval notation.
Example 10:
A ball is thrown vertically from a rooftop
240 feet high with an initial velocity of 64 feet per second.
During which time period will the ball’s height exceed that
of the rooftop? (Use
00
2
16)( stvtth ++−= where
0
v = initial
velocity,
0
s = initial height/position, and t = time. You may
also want to graph this function on your calculator using the
viewing rectangle [0, 10, 1] by [-100, 500, 100]).
Page 6 (Section 1.7)
1.7 Homework Problems
Solve each of the inequalities below and write the answer using interval notation.
1. )31(253 xx
+
≤
+
2. )5(4)2(3
+
≤
+
xx 3.
3
)7(2
2
)3(3
+
<
+
xx
4. )5(
2
3
3
2
−−≤− xxx 5. )1(
2
1
2
1
3
2
4
1
++>−+ xxxx 6. 25232
+
≤
−
<
+
xxx
7. 4
2
3
0 <
+
≤
x
8. 082
2
>−− xx 9. 032
2
≤−+ xx
10. 44
23
+≤+ xxx 11.
23
9xx ≥ 12. 0842
23
≤+−− xxx
13. 0169
2
<+− xx 14. 4
2
<
x
15. 0
3
4
>
+
−
x
x
16. 0
4
3
24
≤
+
−
x
x
17. 0
4
2
≥
−
+
−
x
x
18. 2
3
1
≤
+
+
x
x
19. 5
2
3
≥
−
x
20. 01
1
2
≥−
−
x
x
1.7 Homework Answers:
1.
[
)
∞,1
2.
[
)
∞− ,14
3.
∞−
5
1
,
4.
∞−
7
45
,
5.
−∞−
7
12
,
6. ),2(
∞
7. [-3, 5) 8. ),4()2,(
∞
∪
−
−∞
9.
− 1,
2
3
10. ]2,1[]2,(
−
∪
−
−∞
11. {0}
[
)
∞∪ ,9
12.
{
}
2]2,( ∪−−∞
13.
φ
14.
∞∪−∞ ,
2
1
)0,(
15.
(
)
(
)
∞∪−∞− ,43,
16.
[
)
∞∪
−∞− ,2
3
4
,
17. [2,4) 18. ),3(]5,(
∞
−
∪
−
−∞
19.
5
13
,2
20.
1,
2
1
Page 1 (Section 1.8)
1.8 Absolute Value
In this section you will learn to:
• solve absolute value equations
• solve absolute value inequalities
• apply absolute inequalities
The absolute value of a real number x is denoted by
x
, and is defined as follows:
If
0
≥
x (non-negative number), then
x
= x.
If 0
<
x (negative number), then
x
= -x.
Absolute Value Equations:
If 0
≥
k and
x
= k, then x = k or x = -k.
Solving Absolute Value Equations:
Example 1:
Solve:
732 =+x
Example 2:
Solve:
31
5
32 −=−−−
x
Example 3:
Solve:
261 =++x
Example 4:
Solve:
2143716 ++= x
Example 5:
Solve:
43113
2
=−−− xx
Page 2 (Section 1.8)
Example 6:
Solve (using guess and check) and graph each of the following:
2=x
2>x
2<x
Which solutions involve “and”? Which use the concept of “or”?
Absolute Value Inequalities:
If
,0
>
k then
kx ≥
is equivalent to kx
≥
or
kx
−
≤
(union).
If ,0
>
k then
kx ≤
is equivalent to kx
≤
and
kx
−
≥
(intersection).
Solving Absolute Value Inequalities:
Example 7:
Solve and graph the inequality below.
Example 8:
Solve and graph the inequality below.
Write answer in
interval notation.
Write answer in
interval notation.
345 <−x
3
2
5 +≤
x
Example 9:
Solve:
8213 ≤+−− x
Example 10:
Solve:
520 ≤+< x
Page 3 (Section 1.8)
Example 11:
The temperatures on a summer day
satisfy the inequality
°≤°− 1074t
, where t is the
temperature in degrees Fahrenheit. Express this
range without using absolute value symbols.
Example 12:
A weight attached to a spring hangs at
rest a distance of x inches off of the ground. If the
weight is pulled down (stretched) a distance of L inches
and released, the weight begins to bounce and its distance
d off of the ground at any time satisfies the inequality
Lxd ≤−
. If x equals 4 inches and the spring is stretched
3 inches and released, solve the inequality to find the
range of distances from the ground the weight will oscillate.
Page 4 (Section 1.8)
1.8 Homework Problems:
Solve each of the absolute value equations below:
1.
1285 =+−x
2.
1342 −=++− x
3.
12
11
12
7
6
5
3
2
=−+x
4.
132725 −=+−− m
5. 10252
2
=−− xx
Solve each inequality. Write the solution using interval notation.
6.
8543 <++x
7.
1253 −>−− x
8. 1
4
23
−≤
−
+x
9.
1
5
323 −−−≤−
x
10.
71234 <+− x
11.
3
32
1
−
−
>−
x
12. A Steinway piano should be placed in room where the relative humidity h is between 38% and 72%.
Express this range with an inequality containing an absolute value.
13. The optimal depths d (in feet) for catching a certain type of fish satisfy the inequality
0140035028 <−−d
. Find the range of depths that offer the best fishing.
1.8 Homework Answers:
1. {1, 9} 2. {-6, -2} 3. {-3.5, 1} 4. {2, 5} 5. {-5, -3, 5, 7} 6. (-5, -3)
7. (1, 9) 8.
∞∪−−∞ ,
3
2
]2,(
9. [10, 20] 10.
φ
11. ),3()0,(
∞
∪
−∞
12.
1755 <−h
13. (300,400)
