Transcript College Algebra Week 2

College Algebra Week 2
Chapter 2
The University of Phoenix
Inst. John Ensworth
This week we move on to
LINEAR EQUATIONS
• What is a linear equation?
• Why is it scary?
• It just means you have ONE variable in it,
and that variable isn’t squared or cubed or
anything crazy.
This kind of equation is your friend!
Section 2.1 Page 86
The Linear Equation
• It has one variable : x
ax + b = 0
Where a and b are real numbers and a is not
equal to 0.
A photo album of linear
equations
•
•
•
•
•
•
•
2x+3=0
x+8=0
3x=7
2x+5=9-5x
3+5(x-1)=-7+x
3=8x+5
Etc.
(say cheese!)
How to legally play with linear
equations.
• Most of the magic we work with are really
obvious tricks.
– Remember what happens if we multiply by
things that equal 1? e.g. 5/5 10/10 ?
• Adding the SAME THING to both sides of
an equation does not change the equation…
The Addition Property of
Equality
• So if you start with a = b
• You don’t change anything if you take a
new number or variable and add it to both
sides!
• a+c = b+c
The GOAL
• We want x all by itself on one side and the
rest of the junk on the other (so we can use
our calculator or fingers).
• The technical term for this is: solving for x
EXAMPLE 1 pg 86
• Solve x-3= -7
x–3=-7
x-3 + 3 = -7 + 3
x+0=-4
x= -4
The theme
• ISOLATE THE ‘X’ !!!
Then you can check your answer
• We just got the answer x = -4  so plug it
into the first equation:
• x- 3 = -7
• -4-3 = -7
• -7 = -7 CHECK!
Side note: {-4} is
solution set to the
equation.
** Ex 7-14**
EXAMPLE 2 pg 87
• Solve 9 + x = -2
9+x=-2
9 + x – 9 = -2 –9
x + 0 = -11
x = -11
Solution set {-11}
Checking… 9 – 11 = -2 = -2 CHECK!
Lifesaver: DON’T FORGET THAT LAST STEP!
** Ex 15-24**
EXAMPLE 3 pg 88
• Solve 1/2 = - 1/4 + y
(who cares if the variable is x or y or c or
whatever!)
1/2 = - 1/4 + y
1/2 + ¼ = -1/4 + y + ¼
2/4 + 1/4 = y + 0
3/4 = y
S.S. {3/4}
Checking ½ = - ¼ + ¾ = 2/4 = ½ = ½ check!
** Ex 25-32**
The next trick…multiplication
• That is nice if the stuff we want to move to
the non-x side is just added or subtracted
from the ‘x’, but what if the number is a
coefficient of ‘x’? What if it looks like
3x= 9 ?!
The multiplication property of
equality
• If, again, you have a equation like
a=b
And you multiply both sides by c, then you
haven’t changed the equation!
ac=bc
Again, c can’t be 0.
EXAMPLE 4 pg 89
• Solve x/2 = 6
Note, you can’t subtract or add anything to
get x by itself.
So multiply both sides by 2! (Which is the
INVERSE of the offending number).
2*x/2 = 2*6
2/2 * x = 12
x = 12 SS = {12}
** Ex 33-40**
A quick check…
• Plugging 12 into x/2 = 6
12/2 = 6 = 6 good thing!
Example 5 pg 89
• -5 w = 30
• -5 is in the way!
-1/5 is the inverse of it!
(-5w)/ (-5) = 30 / (-5)
(-5/-5) w = -6
w = -6 The solution set is then
{-6}
** Ex 41-50**
EXAMPLE 6 page 90
• Fractions as coefficients aren’t a problem
either!
• 2/3 p = 40
• If 2/3rd offends, kill it with 3/2nd
(3/2) * 2/3 p = 40 * (3/2)
1 * p = (40*3)/2 = 20*3 = 60
p = 60 our solution set is {60}
** Ex 51-58**
Example 7 page 90
• Are we bothered by –1 as a coefficient?
• Certainly not!
- h = 12
The inverse of –1 is 1/-1 = -1
Multiply both sides by –1!
(-1) –h = (-1) 12
h = -12 SS = {-12}
** Ex 59-66**
Example 8 page 91
Sometimes things are messier…
• What if you have a more goofed up equation?
-9 + 6y = 7 y
There are dumb y’s on both sides… so first we need
to get the y’s on ONE side, THEN do what is
needed to get y alone.
-9 + 6y –6y = 7y – 6y
-9 = y !
Wow, the solution set is there … {-9}
** Ex 67-74**
Example 9
•
page 91
Just one more for good measure:
2/3p =8
(3/2) 2/3p =8 (3/2)
p= 24/2
p= 12
The solution set is … {12}
** Ex 67-74**
Set to Exercises 2.1
• Go to the section you have the hardest time
working with
–
–
–
–
–
–
Definitions Q 1 –6
Solving equations with + and - Q 7 – 32
Solving equations with * and / Q 33 – 54
Doing it with –1 in play Q 55 – 66
Solving it with x on both sides Q 67 – 74
Random mix of anything Q 75 - 94
Section 2.2
• Now we graduate to equations that need
both addition/subtraction and
multiplication/division in the same solution.
• Not a problem… right?
Rules for Happiness
• First do whatever adding and subtracting
you can do…
• THEN do whatever multiplication and
division you can do.
• THEN you should be done!
Example 1 page 95
• Solve 3r – 5 = 0
• Step one… addition or subtraction?
3r –5 + 5 = 0 + 5
3r = 5
• Step two … division or multiplication?
(1/3) 3r = (1/3) 5
r = 5/3
Our solution set is {5/3}
** Ex 5-10**
Example 2 page 95
• Solve -2/3 x + 8 = 0
-2/3 x + 8 – 8 = 0 – 8
-2/3 x = - 8
Mult/Div.: (-3/2) –2/3 x = (-3/2) (–8)
1*x = (-3)(-8)/2
x = (-3)(-4) = 12
Our solution set is {12}
Checking: (-2/3) (12) + 8 = -24/3 + 8 = -8 + 8 = 0 check
** Ex 11-18**
Add/Subt. :
Example 3 page 96
Now with x’s elsewhere and +&- ,*&/
• Solve 3w –8 = 7w
• Group the variable’s
3w –8 –3w = 7w –3w
-8 = 4w
• What’s needed? Get ride of that 4 sticking to the w!
-8 (1/4) = (1/4) 4w
-8/4 = w
-2 = w
Our Solution set = {-2}
** Ex 19-26**
Example 4 page 96
• Solve ½ b – 8 = 12
• Add 8 to both sides first
½ b – 8 + 8 = 12 + 8
½ b = 20
• THEN multiply both sides by the inverse of ½ .
Which is 2!
2* ½ b = 2 * 20
b = 40
Our solution set is {40}
CHECK it… ½ (40 ) – 8 = 20 – 8 =12 = 12 CHECK!
** Ex 27-34**
Example 5 pg 97 - a tad bit more complex
•
•
•
•
The form NOW is ax+b=cx+d
No big change…
Addition first and get the x’s together then proceed
Solve 2m-4=4m-10
2m -4m –4 = 4m –4m –10
-2m -4 = -10
-2m -4 +4 = -10 +4
-2m = -6
(- ½ ) –2m = (- ½ ) (-6)
m=3
Our solution set is {3}
** Ex 35-42**
Checking it…
•
•
•
•
Is 2m-4=4m-10 and
2(3) –4 = 4(3) –10
6 –4 = 12 –10
2 = 2 YES!
m=3
right?
Stepping back in time…
Example 6 page 98
• Remember expanding stuff? (Chapter 1)
You MAY need to do it FIRST, then solve for
x.
We’ll work with 2(q-3) +5q = 8(q-1)
On the next slide…
Workin’ on the railroad
2(q-3) +5q = 8(q-1)
First multiply the parenthesis out, then do the normal
stuff…
2q-2(3) + 5q = 8q –1(8)
2q –6 + 5q = 8q –8
Combine the q’s on the right : 7q –6 = 8q –8
Then subtract 7q from both sides
7q –7q –6 = 8q – 7q – 8
-6 = q –8
Then add 8 to both sides -6 +8 = q –8 +8 give us
2 = q Our solution set is {2}
Don’t forget to check it later! ** Ex 43-50**
Cook Booking It
Good for your note cards!
1. Remove parentheses and combine like terms on
each side
2. Use addition property to get like terms on the
same side as each other (things with x on one
side, numbers only on the other)
3. Use the multiplication property to get the x
alone
4. If x is negative, use the –1 multiplication trick to
make x positive
5. Check your solution to see if you got it!
Section 2.2 Exercises
• Do the hardest first!
–
–
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–
–
Definitions Q 1-4
Simple one variable solutions Q 5 – 18
One variable but on both sides Q 19- 42
Multiply out first… Q 43 – 50
Numbers and variables on both sides Q51-90
Section 2.3 More interesting
equations
• Well, they can have fractions:
Example 1 pg 102
y/2 –1 = y/3 +1
Looking ahead, we can see we need
to get rid of a 2 and a 3 on the bottom.
Multiply the two to get the best number!
6(y/2-1) = 6(y/3+1)
6(y/2) –6 = 6(y/3)+6
3y-6 = 2y+6
3y-6+6=2y+6+6
3y=2y+12
3y-2y=2y-2y+12
y=12
Example 1 the check
•
•
•
•
Plug 12 into y/2-1=y/3+1
12/2 -1 =? 12/3 +1
6-1 =? 4+1
5=5 Check!
** Ex 7-24**
Decimal Fractions
Yes, they can come this way
Example 2 – page 103
0.3p+8.04=12.6
The simplest thing is to kill the decimal
places. The smallest is 1/100ths so we’ll
multiply EVERYTHING by 100
100(0.3p+8.04) = 100(12.6)
Example 2 continues
100(0.3p) + 100(8.04)= 100(12.6)
30p +804 = 1260
30p +804-804 =1260-804
(1/30) 30p = (1/30) 456
p = 15.2
Checking… 0.3(15.2)+8.04 =? 12.6
4.56+8.08 =? 12.6 = 12.6 Check!
** Ex 25-34**
Example 3 page 103
• Or they can be decimal fractions…
(ex 3)
•
•
•
•
•
•
•
•
0.5x+0.4(x+20)=13.4
0.5x + 0.4x +8= 13.4
Let’s get ride of the 10ths by multiplying by 10
10(0.5x+0.4x+8) = 10(13.4)
5x+4x+80 = 134
9x+80-80=134-80
9x=54 Then divide by 9 (1/9)9x= (1/9)54
x=6
Example 3 check…
0.5x+0.4(x+20)=13.4
When x=6
0.5(6)+0.4(6+20)=?13.4
3+.4(26)=?13.4
3+10.4 =?13.4
13.4 = 13.4 Check!
** Ex 35-38**
Some Quick Ones – Ex 4 page 104
a)
2a-3=0
2a-3+3=0+3
2a =3
(1/2)2a= (1/2)3
a=3/2 or {3/2}
And always check!
** Ex 39-54**
b)
2k+5=3k+1
2k-2k+5=3k-2k+1
5=k+1
5-1=k+1-1
4=k or {4}
Equations with more than one
simple answer
(or no answer)
• Identities = stupid equations in which all
numbers are the answers
x/2 = ½ x
x=x
x+x = 2x
3x + 3 = 3(x+1)
x+1 = x+1
yadda yadda
5/x = 5/x (note x can’t equal 0)
Is it an identity? Example 5 pg 105
• Simplify both sides…by themselves
7-5(x-6)+4=3-2(x-5)-3x+28
Kill the ()’s 7-5x+30+4=3-2x+10-3x+28
Group like stuff -5x+41 = -5x+41 Hurray!
** Ex 55-56**
Another definition
• A Conditional Equation
• Is ANY equation that has at least one real
answer (or more) but is NOT an identity
• Everything we’ve done so far is an example
of a Conditional Equation
• Another example x2=4 2 and –2 both
work. (Solution set = {2,-2} )
And another definition
• Inconsistent Equations
• They have NO answer.
x=x+1  x-x = x-x+1  0 = 1 Woa! Yikes!
0 * x + 6 = 7  0 + 6 = 7  6 = 7 Buzz.
Etc.
Finding an inconsistent equation
• Example 6 page 106
Solve 2-3(x-4)=4(x-7)-7x
2- 3x+12 = 4x –28 –7x
-3x + 14 = -3x –28 Hey! That can’t be!
one more step -3x +3x + 14 = -3x +3x –28
14 = -28 Nope, it’s just off.
INCONSISTENT ** Ex 57-74**
Practice section 2.3!
• Do it do it do it!
– Definitions Q 1-6
– Looking for conditional equations, inconsistent
equations or identities Q 7 – 24
– Decimals Q 25-38
– Good practice Q 39-54
– Anything goes mixed bag Q 55-92+
On to Section 2.4 Formulas
• Don’t be fooled, this isn’t any harder except
we sometimes don’t have ANY obvious
numbers… we have letters and we choose
one variable to be OUR variable for
solving.
• You just leave the answer with the letters
however they fall… but the tricks are all the
same as what we have already used.
The trick…
• Pretend all the OTHER variables are just
numbers to get out of the way.
Definition
• A formula or literal equation is an equation
involving two or more variables. We solve
it for a certain variable.
• Making formulae work for us is a main
reason to even worry about solving for any
particular variable! This can be useful!
Example 1 page 110
• Solve D=RT for T…
• D/R = (R*T)/R divide both sides by R to get T
alone
• D/R = T
• Swap sides  T = D/R
done!
** Ex 7-18**
Example 2 page 111
•
•
•
•
•
•
•
•
The Celsius to Fahrenheit equation…
C = 5/9 ( F –32) for F
We need to get rid of the 5/9 with 9/5
(9/5) C = (9/5) (5/9) (F-32)
(9/5) C = F-32 then add 32 to both sides
9/5 C + 32 = F (-32 + 32)
9/5 C + 32 = F switch sides
F= 9/5 C + 32 done!
** Ex 19-24**
Example 3 page 111
• Solving for x when it’s on both sides
(you’ve already done this) but now you
have some other letters going along for the
ride like numbers.
Example 3
•
•
•
•
•
•
5x – b = 3x + d subtract 3x from both sides
5x –3x -b = d  2x –b = d
Add b to both sides
2x = d + b
Divide both sides by 2
x= (d+b)/2
done!
** Ex 25-32**
Note on speed…
• Note, I’m dropping some steps in solving
equations that you should be getting
smoother at. I’m labeling the steps, but not
showing ever tiny part.
Example 4 page 112
•
•
•
•
•
x + 2y = 6 Solve it for y
Subtract x from both sides 2y = 6 – x
Multiply both sides by ½
( ½ ) 2y = ( ½ ) ( 6-x)
y= (6-x)/2
is good BUT you can also do it thus…
• y= ½(6-x) = 6/2 –x/2 = 3-x/2 = -x/2 + 3
** Ex 33-42**
Example 5
page 112
• Solve 2x-3y = 9 but make it look like y=mx+b
(the slope of a line, to be used a LOT later!). m
and b are what ever numbers they turn out to be.
• We want y alone on the left… so subtract 2x from
both sides.
• -3y = 9-2x Then divide both sides by –3
• y = 9/(-3) – 2x/(-3)
• y= -3 + (2/3)x
then swap the number and the x
• y= (2/3)x –3 looks like y= mx+b m=2/3, b=-3
** Ex 43-54**
Finding the value once you’ve
solved the equation Ex 6 pg 113
• In example 5 we found that 2x-3y-9
becomes y= (2/3)x –3
• If we’re told that x=6, what is y?
• y= (2/3) 6 –3 = 12/3 – 3 = 4 –3 = 1
• That means on this line (more on that later)
if x = 6 then y = 1.
** Ex 55-64**
Example 7 The simple interest
formula page 114
Solve I=Prt You know the simple interest $120 (I),
the principle is $400 P) over 2 years (t). What is
the rate (r) ?
Prt = I divide both sides by P and t
Prt/(Pt) = I/ (Pt)
r= I/Pt = 120/(400*2) = 0.15 or 15%
fun no?
** Ex 73-76**
Example 8 page 114
• How about using the perimeter equation for a
rectangle? What is the Length (L) if the Perimeter
(P) is 36 feet and the width (W) is 6 feet?
• P=2L+2W
• P-2W= 2L
• 2L= P-2W divide by 2
• L= (P-2W)/2 = (36-2(6))/2 = 12 feet
** Ex 77-80**
Section 2.4 Formula Practice
• Give ‘em a test drive…
–
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–
–
Definitions again Q1 – Q 6
Working with formulae Q 7 – 18
Letters instead of numbers in equations Q 19-24
Making things look like y=mx+b Q 25- 32
The same thing but with x=2 Q 33-64
Tables of numbers Q65-72
Working with word problems of all the above
• Q 73-101 USEFUL!
Section 2.5 – Putting this into
words
• This section expands on the verbal work
you did in 1.6
Verbal Phrases… Math as a
language again… Addition first
• Addition:
–
–
–
–
The sum of a number and 8
Five is added to a number
Two more than a number
A number is increased by 3
x+8
x+5
x+2
x+3
More talk - Subtraction
• Subtraction
–
–
–
–
Four is subtracted from a number
x-4
Three is less than a number
x-3
The difference between 7 and a number x-7
A number is decreased by 2
x-2
Multiplication
• Multiplication
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–
–
–
The product of 5 and a number
Twice a number
One half a number
Five percent a number
5x
2x
½x
0.05x
Division
• Division
– The ratio of a number to 6
– The quotient of 5 and a number
– Three divided by some number
x/6
5/x
3/x
Algebraic Expressions in Words
• Example 1 page 120
–
–
–
–
–
The sum of a number and 9
x+9
Eighty percent of a number
0.80w
A number divided by 4
y/4
The result of a number subtracted from 5 5-z
Three less than a number
a-3
** Ex 7-18**
Adding wording for
pairs of numbers
• If you sort of know how two numbers are
related to one another, then you can
construct one of those equations we’ve
worked with that has two x’s in it.
• Like addresses… “that house is three
houses down the street from this house.”
And I know the address of this house, so I
can figure out the other house’s number.
I know…
• I have two numbers… one number is 10
less than the other…
• I can write the first number as x
• And the other is 10 more than it, then it is
x+10
Example 2 page 121
• Write two numbers that differ by 12
x and x-12
Check x – (x-12) = x-x+12 = 0 +12
• Two numbers with a sum of –8
x and -8-x
Check x+ (-8-x) = x-x-8 = 0 –8 = -8
The trick, write down x. Write down the answer +
the opposite of the variable x (-x). Then show if
you are adding or subtracting the values.
** Ex 19-28**
Example 3
A related subject
page 122
• If you know the overall angle in some
geometric shape…you can do this same
‘two things added’ verbal trick.
• But now we know the overall angle!
Ex 3 continued
• See page 121… a)
• The right angle has 90 degrees. Fact.
• A smaller angle that is x and is part of the
overall 90 degree angle. The rest of the
stuff in that 90 degree angle is ?. So the
remainder is 90 – x or ? = 90-x
• Nutty simple? If not, you are overthinking!
Ex 3 b and c
• (b) So if you have in (b) 180 degrees and a
smaller part of that called x. Then the rest of the
stuff in the 180 degree angle is 180-x
• (c) The interior angles of all triangles equals 180
degrees. So if you know one angle (30 degrees)
and another angle is x, then the rest of it is: 180 –
x –30 or 150 – x
** Ex 29-32**
Yet another in the same vein…
•
Example 4 page 123 describing sets of numbers
a) Write three consecutive integers, the smallest
of which is w
 w, w+1, w+2
b) Write three consecutive even integers, the
smallest of which is z
 z, z+2, z+4
4c
• c) Four consecutive odd integers, the
smallest of which is y
y, y+2, y+4, and y+6
** Ex 33-40**
MORE Writing Ex 5 pg123
• How about using known formulae?
– a)
– The distance if the rate is 30mph and the time is
T hours
– We go find the formula D=RT for
distance=rate times time
– Plug in the known value… R=30mph
– So D=30T
done!
Example 5b
• The discount rate if the rate is 40% and the
original price is p dollars
• We go get the formula
Discount = Rate of discount * Original
Price
• Since the discount is the rate times the
original price, an algebraic expression for
the discount = 0.40p dollars
** Ex 41-64**
Page 124 is your repository of
equations for these problems!
• See this page for many normal, everyday
formulae.
• Use these for the quizzes and all!
• Note cards again?
Example 6 Words and Formulae
page 124
• a) Find two number that have a sum of 14
and a product of 45
• First, x is one of the numbers. Just say it.
• The other number is 14-x (the answer and
the opposite of the first number)
• Their product is 45 : x(14-x) =45 done!
Ex 6b
• The coat is on sale for 25% off the list price. If
the sale price is $87, then what is the list price?
• Our variables. x is the original price (just say it)
The amount of discount is 0.25x
• Which formula on pg 97?
Original price – discount = selling price (the third
one)
• Plug it in: x –0.25x = 87 done!
Example 6c
• What percent of 8 is 2?
• If x is the percentage, then we want to
know the ratio 2/8
• x=2/8 or 8x = 2
• Don’t forget to multiply the answer by 100
to make it a percent in the end.
Example 6d
• The value of x dimes and x-3 quarters is $2.05
• We’ll say the value of x dimes at 10 cents each is
10x cents.
• The value of x-3 quarters at 25 cents each is 25(x3)
• The sum of them is 10x+25(x-3) = 205
(the $2.05 was turned into cents as well! Unit must
match!)
** Ex 65-90**
Work on Section 2.5 Questions
for a bit!
• Go for the hardest first and get help!
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–
–
Definitions Q1 to Q6
Translating words to equations Q 7-18
Pairs of numbers Q19-27
Playing with the same idea but with angles Q29-32
Words describing sets of numbers to expressions
Q 33 to 40
– Words describing expressions to expressions Q 41-64
– Words describing equations to equations Q65-124
Let’s do some real world stuff
before getting to something
called inequalities…
Now that you can turn sentences into
equalities, let’s just do some ‘real world’
type problems.
2.6 Applications
Example 1 Number Problems page 130
• The sum of three consecutive integers is 48.
What are they?
• They are: x, x+1, x+2
• x+(x+1)+(x+2)= 48 regroup
• 3x+3=48
subtract 3 from both side
• 3x=45
• x= 45/3 =15 So they are 15,16,17 (plug in)
** Ex 7-14**
Cookbook for Word Problems
1. Read the problems as many times as necessary.
Guessing the answer and check to see if you can
just ‘get’ it.
2. Draw a picture if you can.
3. Choose a variable (i.e. x) write what it represents.
4. Write the small algebraic expressions for
unknown stuff.
5. Write an equation that puts it all together.
6. Solve the equation.
7. Answer the original question (plug in the answer)
8. Check your answer!
Try another perimeter problem
Example 2 page 131
• If the length or a property is 1 foot less than
twice its width and the perimeter is 748 feet,
find the width and length.
• x is the width, we say.
• The length is then 2x-1
• The equation is 2L+2W=P always
• Plug it in: 2(2x-1)+2x=748 then solve…
Ex 2 goes on…
• 2(2x-1)+2x=748 multiply out stuff
• 4x-2+2x=748 collect like terms
• 6x-2=748
add 2 to both sides
• 6x=748+2 = 750 divide by 6
• x= 125 But We’re NOT Done Yet!!!
• Width = x Length = 2x-1
• Width = 125 feet Length 249 feet
THEN CHECK YOUR ANSWER ** Ex 15-20**
A bit’o’geometry Ex 3 page 131
• (See figure on page 132)
• The angle formed by the guy wire and the
ground is 3.5 times as large as the angle
formed by the guy wire and the antenna.
Find the degree measurement for all the
angles.
Ex 3 continued
• We’ll say x is the degree of measure of the
smaller angle
• So the larger is 3.5x
• The third angle is 90 degrees since it is a
right angle
• If you have a right triangle, the other two
angles always add to 90 degrees
(You just need to know this or look it up.)
Doing Ex 3
• Add all 3 of ‘our’ angles
x+3.5x=90 add like terms
4.5x = 90 divide both sides by 4.5
x= 90/4.5 = 20
So one angle is x = 20 degrees
The other is 3.5x = 3.5(20) = 70 degrees
DO they add to 90? 70+20=90 check!
** 21-22**
REALLY common time and rate
problems
• Billy rides the bus for 3 miles at 40 mph
then the bus accelerates to 70 mph for 5
hours. The entire trip covered 400 miles.
• What color was Billy’s hat?
Example 4 page 132
• Car goes 2 hours on an icy road.
• On dry road, speed increases to 35mph for 3
hours.
• Total trip 255 miles.
• How many miles of icy road?
 Make a Rate, Time, Distance Chart and fill in
what you know, and put variables on what you
don’t…
Ex 4 continued
Rate
D=RT
Time
Distance
Icy Road x mi/hr 2 hr
2x mi
Clear
Road
3(x+35)
mi
x+35
mi/hr
3 hr
Since we know the total distance 255, we’ll work with
the distances we have. Add ‘em up
2x+3(x+35)=255 multiply out the ()’s
2x+3x+105=255 simplify and subtract 105 from both sides
5x=150 then divide by 5 x=30 Which is the answer 30mph.
** 23-26**
Example 5 page 133
• Driving average 55mph
• Trip back (same route) took 3 hours longer
at 40mph
• How long for first trip?
• Distance start to end?
Ex 5 continued
D=RT
Rate
Time
Distance
Going
55 mph
x hr
55x mi
Coming
40 mph
x+3 hr
40(x+3)
mi
Since we know the total distance = going + coming
55x=40(x+3)
eliminate ()’s
55x=40x+120 subtract 40x from both sides
15x=120
divide both sides by 15
so x=8 Going time = 8hours, Distance =55(8)=440miles
** 27-28**
Work on Section 2.6
Questions for a bit!
• Go for the hardest first and get help!
– Word problems galore Q1-44
Section 2.7 Word problems about
finances
Discount Problems:
Example 1 page 138
•
•
•
•
•
•
•
A 12% discount
$6606 discount
Original Price?  x
Use equation: Rate of discount ·Price=Discount
0.12x = 6606
Plug in knowns leave x
x=6606/0.12 calculator!
x=$55,050 Done! (check your answer)
** 7-8**
Example 2 page 138
• Discount rate 12%
• Sale price of car $17,600 (discount)
• Original price? x
Use:
price-discount=sale price
x-0.12x=17,600 combine like terms (with x’s)
0.88x=17,600 divide by 0.88
x=17,600/0.88
x=$20,000 was the original price!
** 9-10**
Commission Problems
Example 3 page 138
•
•
•
•
•
•
•
Commission rate is 7%
Selling price?  x
So can get $83,700 take home
Selling price – commission = take home share
x – 0.07x = 83,700
0.93x = 83,700
x=83,700/0.93 = $90,000
** 11-14**
Investment Problems
Example 4 page 139
• Best to make your table!
Interest
Rate
Amount
Invested
Interest
for 1 year
CD
9%
x
0.09x
Mutual
Fund
10%
2x
0.10(2x)
CD Interest + mutual fund = total interest
0.09x+0.10(2x) = 232 now we’ll solve it!
Ex 4 continued
0.09x+0.10(2x) = 232
0.09x+0.20x=232
0.29x =232
x= 232/0.29
x=$800  was invested in the CD
2x=$1600  was invested in the Mutual Fund
** 15-18**
Work on Section 2.7
Questions to get better!
• Go for the hardest first and get help!
– Word problems galore Q1-42
Section 2.8 All the same things
with inequalities instead of =
signs
• It’s a shift in what ‘equals’ means.
• This allows you to practice all the tricks and
games you’ve learned so far.
• Remember the number line?
– Is
–5 greater or less than 4?
• But it starts with some new definitions…
2.8 Inequalities
• The Basics
•
•
•
•
< Is less than
=! Is not equal to
<= Is less than or equal
> Is greater than
>= Is greater than or equal to
We are comparing numbers
(or sides of an expression)
• We are comparing them then asking if it is
true or not (much of the time).
• True or false? -10 < -5
Example 1 page 145
•
•
•
•
•
•
•
Draw a number line if you need to (from last week)…
a ) is 3<4 ? True
b) is -1< -2 ? False
c) is -2 <= 0 ? True
d) is 0 >= 0 ? True
e) is 2(-3)+8 > 9? -6+8>?9 2>?9 False
f) is (-2)(-5)<= 10? 10<=?10 True
** Ex 7-22**
We can mark it on the number
line!
• Remember the solution set? We put the
answer in { }’s
• Any solution set can be written on the
number line
• For instance, All x’s such that x is less than
three… {x|x<3}
Note the “ ) “
Another example
• This is the number line graph all x’s such
that x is greater OR equal to 1.
• {x|x>=1} Note the “ [ “ this time for the
equal
Watch the (‘s and [‘s and the way
they point.
The basic number line again
----|----|----|----|----|----|----|----|----|----|----|----5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2
• a)
• x<5
< --|----|----|----|----|----|----|----|----|----|----)---5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2b
• b)
• -2<x
---|----|----|----(----|----|----|----|----|----|----|--->
-5 -4 -3 -2 -1 0 1 2 3 4 5
Example 2c
• c)
• x>=10
---|----|----|----|----|----|----|----[---|----|----|--->
2 3 4 5 6 8 9 10 11 12 13
** Ex 23-34**
Example 3 page 146 –
More Graphing
a) 2<x<3
----|----|----|----|----|----|----|----(----)----|----|----5 -4 -3 -2 -1 0 1 2 3 4 5
b) –2<= x <1
----|----|----|----[----|----|----)----|----|----|----|----5 -4 -3 -2 -1 0 1 2 3 4 5
** Ex 35-42**
We checked equalities before,
now we’ll do inequalities!
Ex4 pg147
a) 2x-3<=? –5 if x=0
2(0) –3 <=? –5
0-3 <=? –5
-3 <=? –5 no. False.
b) x-5 > 2x+1 if x=-4
-4-5 >? 2(-4)+1
-9 >? -8+1
-9 >? -7 no. False
Example 4 continued
c) 6 < 3x-5 <14 if x=13/3
6 <? 3(13/3) –5 <? 14 (the 3’s cancel)
6 <? 13 –5 <? 14
6<? 9 <? 14
Yes! True.
** Ex 53-70**
Turning Words into Actions
Example 5 page 148
a) Lois plans to spend at most $500 on a
washing machine including the 9% sales
tax. Write the inequality…
The total must be less than $500
The washing machine costs x, the tax is 0.09x
So x+0.09x <= 500
More real world
b) The length of a certain rectangle must be 4
meters longer than the width, and the
perimeter must be at least 120 meters.
We’ll call the width W, so the length is W+4
Looking up the perimeter equation from
somewhere 120<= 2W +2L
So that gives us 2W + 2(W+4) >= 120
And one more:
c) Fred made a 76 on the midterm exam. To
get a B, the average of his midterm and his
final exam must be between 80 and 90.
If we call his final score x … is average is
(x+76)/2
so that is between 80 and 90
80< (x+76/2) <90
Practice for 2.8
Go for the hardest first and get help!
–
–
–
–
–
–
–
Definitions Q1 to Q6
Inequalities Q 7 to 22
Graphing Inequalities Q 23 to 34
Graphing Compound Inequalities Q 35 to 42
Write the inequality from a graph Q 43 to 52
Checking inequalities for truth Q 53 to 78
Writing inequalities plus Q 79 to 91
2.9 Solving Linear Inequalities
The official properties…
• If you multiply both sides of an inequality by a
negative number you must switch the inequality sign
In SIMPLE terms
• All the rules are the same as you worked
with equalities EXCEPT if you multiply or
divide by a negative number
• Multiply or divide by a negative number
and you switch the > to a < or the reverse.
2.9 Example 1 page 154
Writing in Inequalities (a sneaky way to get
you to solve them without knowing it)
a) x+3 >9, x ___ 6 (fill in the blank)
subtract 3 from both sides
x> 9-3
x>6
b) -2x <= 6, x ___ -3
divide both sides by –2 !!! Switches the <> thing!!!!
x >= -3
Example 2 page 154
• 4x-5 >19
• 4x-5+5 > 19+5 !!! Does NOT switch the <>’s !!!
• 4x>24
• x>6
----|----|----|----|----|----|----|----|----|----|----|----(----|--->
-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
** Ex 17-18**
More complex, but JUST like what you’ve been
doing and doing and doing!
Example 3 pg 155
• Nothing new here except when you multiply or divide by a
negative… that’s all!
• 5x-2 <= 7x –5
• 5x-2-5x <= 7x-5-5x
• -2 <= 2x-5
• -2 +5 <= 2x-5+5
• 3<=2x then divide both sides by 2 Positive 2.
• 3/2 <= x
<---|----|----|----|----|----|----|--]--|----|----|----|----5 -4 -3 -2 -1 0 1 2 3 4 5
** Ex 19-22**
Example 4 page 156
Negative!! Aaah!
•
•
•
•
•
•
5-5x <= 1+2(5-x)
5-5x <= 1+10-2x -2x to both sides then add stuff
5-3x <= 11
-5 from both sides
-3x <= 6
divide both sides by –2 warning warning change the < !!!
x>= 6/-3 = -2
x>= -2
----|----|----|----[----|----|----|----|----|----|----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5
** Ex 23-44**
Ex 5 The same thing BUT with 3
parts (compound inequality) pg 156
-9 <= 2x/3 –7 <5 add 7 to all 3 parts
-2 <= 2x/3 < 12 multiple all 3 parts by 3/2
-2(3/2) <= (3/2)2x/3 < (3/2) 12
-3 <= x < 18 we never multiplied or divided by a
negative number, so the <>’s didn’t change.
----|----[----|----|----|----|----|----|----)----|----|----6 -3 0 3 6 9 12 15 18 21 24
** Ex 45-48**
3 parts AND a negative!
Ex 6 pg 157
-3 <= 5-x <= 5
-3 –5 <= 5-x-5 <= 5-5
-8 <= -x <= 0
(-1)(-8) >= (-1)(-x) >= (-1)(0) The Switch!
8>= x => 0
2.9 Exercises and Fun!
•
•
•
•
•
•
•
How do you get to Carnegie Hall?
Do Inequalities…
Definitions Q 1 – Q 6
Fill in the blank Q 7-16
Solving simple ones Q 17 - 44
Solving 3 part-ers Q45- 64
Dem Word problems Q65- 80