_
SoIvin2 One-Step Equations
er
Solution of an equation
__‘Q
kflQ.
Equivalent equations:
k
Properties of Equality:
Let a, b, and c represent real numbers.
AddifionProperty:
CC+C
Ex.
(
Ex.
f\
5hQ
\‘
Subtraction Property:
b, - c-c b
&
-
Multiplication Property:
Ex.
Division Property:
6
--c’
1’
c
Ex.
—1
T
1.
-
H*m
Substitution Prop’rty
f
Ex.
W
pc4 W
Inverse operations:
V
1
\
c\cx
1h(
‘
r\OL
M
2b+3b
O(’& ()Jxiç
Di v scr
Example 1: Solving Using Addition or Subtraction
a. x—3=—8
b.
g+7=l1
] -1
(i)t]j1
Example 2: Solving Using Multiplication or Division
b.
a4.-x=9i
3
Q
L2i
—96=4c
u
CbQci’
tP’) =9
=
q-
‘
C.
‘_z—2.’
__
Section 3.1
Solving Two-Step Equations
Steps to follow for solving two-step equations:
1. ‘Ddc GS cä*cs O srOic
cc c\iScA
2.
Example 1: Solving a Two-Step Equation
m
a. Solve 10=—+2
b.
x
CkQC(’
x—3=2
6
-a
‘1
3=
5-3z
oc
2aJ
Example 2: Real-World Connection
A music store sells a copy of deluxe electric guitar for $295. This is $30 more than -i-the cost of
the deluxe electric guitar it is modeled after. What is the cost of the deluxe electric guitar?
c
C
oroJ Øos o
-
Example 3: Writing a Function
In a catalog, tulips cost $0.75 each and shipping costs are $3.00. Write a rule that describes the
spent as a function of the number of bulbs ordered. Then determine the greatest number
of bulbs that you can order for $14.
amount
5
=
cs*h
bftS
:Thft ±
3
1’2 \CSt \(iJ cQji
bij ff s i9±uIip
¶0
Example 4: Using Deductive Reasoning
a. Solve 1
=
+5 and justify each step.
(sbco of ‘)
—
prep c4
b. Solve 9—b =11 and justify each step.
-9
-1
O—
1is 3OO
(Divjso jiopd
)
Section 3.2
‘—
So1vin Multi-Step Equations
Steps for solving a multi-step equation:
1. Ccr +
cc
2.
O
cy
3.
\\4çQ
.
Example 1: Combining Like Terms
Solve each equation.
a. 2c+e+1278
b. —4b+16—2b=46
—(sb
-U 3ö
Th
Example 2: Real-World Connection
A gardener is planning a rectangular garden area in a community garden. His garden will be
next to an existing 12-ft fence. The gardener has a total of 44 ft of fencing to build the other
three sides of the garden. How long will the garden be if the width is 12 ft?
i ft
i=X oc4.
;x==
Example 3: Solving an Equation with Grouping Symbols
ii
-
a. Solve
—
2(b —4)
=
12
b. Solve —31
=
—(4x —5) + 3x —6
=
-b
F1
3O
Example 4: Solving an Equation That Contains Fractions
2x x
a. Solve —+—=7
I’3 2
14
y
b. Solve 2
—y—----=l
2
5
z
4x
)7(
01
N=L44Q\
Example5: SoIvin an Equation That oiitiiDecima1s
a. Solve O.5a + 8.75
=
13.25
b. Solve —2.3x+8+1.43x=—3.68
=
-ixCO
5cc zç
‘-S
2-.
-3
Section 3.3
Equations With Variables on Both Sides
Example 1: Solving an Equation With Variables on Both Sides
a.
b.
7x—5=2x+15
2(
iN
—(x+6)+5x+8=2—2x
ZE
2-
E
Example 2: Vertical Angles
Find the value of x in the diagram below.
/
‘I
V ±
S
+3E
—
/
-
3=X
ocj
Example 3: Real-World Connection
You can buy used in-line skates from your friend for $40, or you can rent some. Either way, you
must rent safety equipment. How many hours must you skate for the cost of renting and buying
skates to be the same?
SKATE RENTALS
In-linc zkatcz and
stftv cupmcnt
$3.SO/ hour
Sacty cquipmcnt
X 35oX
1
6+[
3cK
t
1
‘
5(X
--
-oc 2 s
Cc4 k()
.
Qi
Example 4: Solving Using a Graphing Calculator
Solve
m =8—
\:
1
-c
m using a graphing calculator.
3’.
(1\
a *c
o\
ck \QkQ
____
Identi:
No solution:
In
ic’±
1c
\- -vu.
Example 4: Identities and Equations With No Solutions
a. Solve 10— 8a
2(5 4a)
—
( \Q
b. Solve 6rn —5
=
7rn+ 7—rn
-qç %C\
—5=1
QQ
Sce Q
c\
0
*\k\ rc\
QQO(Q
\rc)S -çcQ O\O\C\S
__
Section 3.4
Ratio:
:cpco\
Ratio and Proportion
bc ct
Rate:
o
Unit Rate:
C( c
AbQS
riS
1
b
c \id’),-4s
(i
s c
•
mCc1ck( 1’
L
Example 1: Using Unit Rates
The table at the left gives prices for different sizes of the same brand of apple juice. Find the
unit rate (cost per ounce) for the 1 6-oz size.
(Z’
/
:
$
(‘
c
f’
1!-
o
s_i)
Example 2: Real-World Connection
Tn 2004, Lance Armstrong won the Tour de France, completing the 3391 km course in about
83.6 hours. Find_Lance’s unit rate, which is his average speed. Write a rule to describe the
distance he cycles d as a function of the time t he cycles. Cycling at his average speed, about
how long it would take Lance to cycle 185 km?
3C
4i
X”
—
Z1’1
14
J
i4’\
,
Example 3: Converting Rates
—
..i
bJ)
A cheetah ran 300 feet in 2.92 seconds. What was the cheeta1’Thi gespëdin miles per
hour?
_c
c 4
4L
—L-
lc)Mr
-
52?c
F-’
-
C
—fl
Section 3.5
Proportions and Similar Figures
Similarfigures:-GQS Q*
‘nc’JQ- *Q- X1’&
‘c’Q. S)-Q StQ
ThQ sb
Th
occx c
co o
-+
s
‘\J
Q
os’qs
2_-
Example 1: Finding the Length f a Side
AABCADEF.FindDE.
e.pY
Cc
roI
C
MCft-kO s
C” 1\
D
/f
/
//
F
cm
cm
ri)
/
1
C
2
J
7
‘c_ç_
(—i
-
5. cm
z
i
sx
Dilation:
n
(‘
fl
h—-[ir. c’
OQS\co
Scale Factor:
—--
L
c’o-o
I
-h Oc.r-
cc
(‘YiQ *()
_nQ QccS
Example 2: Dilating Figures on the Coordinate Plane
Quadrilateral PQRS has vertices P(2, 4),Q(4, 4), R(4,-2), and S(-4,-4). It is dilated by a scale
factor of, and the origin is the center of dilation. Graph the original figure and its dilation.
:
p)
D
sl
2’1
L)
L2-1)
-q-= (E2-2)
Example 3: Applying Similarity
A tree casts a shadow 7.5 ft long. A woman 5 ft tall casts a shadow 3 ft long. The triangle shown
for the tree and its shadow is similar to the triangle shown for the woman and her shadow. How
tall is the tree?
K
Scale drawing:
EZ
Sc vc’
‘S
cr
Scale:
n
EV’Xt
co
-—
1’\r\( ((* ‘.o.
L
rim.
Example 4: Finding Distanes on Maps
The scale of the map at the left is 1 inchi1O miles. Approximately how far is it from Valkaria to
Wabasso?
yalicaria
4tE/ ‘JT’1(
-
irat
-.
d
Micco
hs&and
\
Sebstian
abasso
ii
/
‘.....
I
••_
E
Section 3.6
Equations and Problem Solvin2
Exampie 1: Defining One Variable in Terms of Another
The length of a rectangle is 6 in. more than its width. The perimeter of the rectangle is 24
in. What is the length of the rectangle?
L4 q
f)= zi
-hhn
j4(
:
.:
J31
,
zyU
>}‘1n.
Consecutive Integers:
r— bu I
;
*
.
f
•7
Example 2: Consecutive Integer Problem
The sum of three consecutive integers is 147. Find the integers.
w*r
3x 3
iJ-
[ft
-
flflD
I
q
Uniform Motion:
rr cbc+ -Hf
-c
cY o
WH\
ci.
+h
d
dfld) r 1)
c
Example 3: Same-Direction Travel
A train leaves a train station at 1 P.M. It travels at an average rate of 72 mi/h. A highspeed train leaves the same station an hour later. It travels at an average rate of 90 mi/h.
The second train follows the same route as the first train on a track parallel to the first. In
how many hours will the second train catch up with the first train?
a. Draw a diagram:
S
b. Define a variable:
(- -
-+hn
t-
mL±
cc-
--ron
ho-
c. Complete the table:
d. Set up an equation and solve:
\JhQfl WW
JQ_
*bc
S cvQ C Sk-cflcQ 7
doe
j(
Ht -9C
Lz5
i
h12_
TtOJ.fl
4 bcs.
Lv:)
I Cod-c op +o +h
-
Section 3.7
Percent of Change:
-
Percent of Change
c&)U Oc
-
p cQc-\
)J
0 i cj noi
Percent of Increase:
f0
O
oo)rt
Percent of Decrease:
ocCs
LTh O \XjXu
coe s
-rcc’
‘
Lc-
oNc)r*
c9r\c
Example 1: Finding Percent of Change
The price of a sweater decreased from $29.99 to $24.49. Find the percent of decrease.
/
99 ) L
-
—
9
bC
—
-______
O\
CC
Example 2: Real-World Connection
In 1990, there were 1330 registered alpacas in the United States. By the summer of 2000,
there were 29,856. What is the percent of increase in registered alpacas?
•Q2i
%J
‘(
i?/\
ç:;
1
_)y
‘..
•
—
Greatest Possible Error:
‘
s
ThQ
•
tZ() Oc
QQ(
\QO
6c O5..
c,
5
___
_____
Example 3: Finding the Greatest Possible Error
You use a beam balance to find the mass of a rock sample for a science lab. You read the
scale as 3.8 g. What is your greatest possible error?
[h
‘S +\
\s
Li.RE,s
Example 4: Finding Maximum and Minimum Areas
You measure a room and make the diagram below. Use the greatest possible error to find
the maximum and minimum possible areas.
Th_
rsy
+COT.
131t
7
ro
+c
s-
is
or
ft
—
0
Percent Error:
cjsà
1
fl
—
-
—
Example 5: Finding Percent Error
Suppose you measure a CD and record its diameter as 12.1 cm. Find the percent error in
your measurement.
rncnQc*
s
)
0
=
—
S kc W
(*)
Example 6: Finding Percent Error in Calculating Volume
The diagram below shows the dimensions of a cassette case. Find the percent error in
calculating its volume.
c. 5°
III.’.) L.I•Il
bPE ‘3
oS
•)çy5o4r
\csu\ed
\J
I
‘
rn
kc
c’q
v
Ttc )
kn Vorr
-
L
F
‘31
S Ci 5
5
-
id
LC]5
c-
•
L°/
Q
QF
Section 3.8
Radical:
Finding and Estimatin2 Square Roots
rQ
rcs
c
H
Radicand:
-
rco+
Qr *\
Perfect Square:
cadc
C+
cmaj4’c
xr
jt c-ri
-.
-
S9r
_
I
Some common perfect squares:
1)
1)iOQ)
2g
Principal Square Root:
4Q
199
(9
R-
4
Negative Square Root:
Q
1hn
flQ9o±t VQ flxfr 1±
hhQ
;
Example 1: Simplifying Square Root Expressions
Simplify each expression.
z
-((N
-
pesrtM S
It
9
ichv
oo1
d.±J?i
•
=0
1
fl. (Q FS
C
e.
S9JGR RXY
0
rot
flQU±rc
VQ
Sinc i±
4€c-rni ncQ
lNumbers:Nl\OS
:
Irrational Numbers:
)
do $
m.
h
J1b(S *hDz\
ifl
O
*ioy cr
..
‘-Y
Example 2: Rational and frrational Square Roots
c._
i
Tell whether each expression is rational or irrational.
a.
49
b.
-
FC\OnoJ
rc*onoJ
.
_3uo-1ci11,
C’.
\/
=
5113O’M9
roonc
Example 3: Estimating Square Roots
Between what two consecutive integers is J14.52?
N
i5
uho
V\JQ
I
±UO
3
Srf
+h
Example 4: Approximating Square Roots With a Calculator
Find J14.52 to the nearest hundredth.
iQçf•
or 3
_
__
Example 5: Real-World Connection
2 gives the length d of each wire for the tower below. Find
The formula d = Jx2 + (2x)
the length of the wire if x = 12 ft.
1
/
+
//
/
)
Section 3.9
The Pythagorean Theorem
Parts of a Right Triangle:
Hypotenuse:
-hQ çV*
c *Q c\Q
Leg:
9 Ofl91Q
r
The Pythagorean Theorem:
In any right triangle, the sum of the square of the lengths of the legs is equal to the
square of the length of the hypotenuse.
1
I
JO
U
Example 1: Using the Pythagorean Theorem
What is the length of the hypotenuse of the triangle below?
q+j
9 ;ni
t÷
(ill
=
Cr\5
Example 2: Real-World Connection
A fire truck parks beside a building such that the base of the ladder is 16 ft from the
building. The fire truck extends its ladder 30 ft as shown below. How high is the top of
the ladder above the ground?
4
clc)G
Conditional:
Hypothesis:
Conclusion:
y
-
t
c cdcncd
th
Converse:
n
(S
n ft
&n\
vcj t
Xsk 4
hQ *hQ
The Converse of the Pythagorean Theorem:
2
If a triangle has sides of lengths a, b, and c, and a
c.
right triangle with hypotenuse of length
2
+b
=
COCSm
, then the triangle is a
2
c
Example 3: Using the Converse of the Pythagorean Theorem
Determine whether the given lengths can be sides of a right triangle.
b. 7m,9m,andl2m
a. 5in.,l2in.,andl3in.
c
)Qfl
)*Qfl
rg\
5’
s s
r
in s
•
S
o
a