College Algebra Week 2Chapter 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 (say cheese!)• x+8=0• 3x=7• 2x+5=9-5x• 3+5(x-1)=-7+x• 3=8x+5• Etc.

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 = - 7x-3 + 3 = -7 + 3

x + 0 = - 4x = - 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 = - 29 + x – 9 = -2 –9

x + 0 = -11x = -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 + y1/2 + ¼ = -1/4 + y + ¼2/4 + 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 = bAnd 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*62/2 * x = 12

x = 12 SS = {12}** Ex 33-40**

A quick check…

• Plugging 12 into x/2 = 612/2 = 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 = 12The 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 andmultiplication/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 + 53r = 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

Add/Subt. : -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) = 12Our solution set is {12}Checking: (-2/3) (12) + 8 = -24/3 + 8 = -8 + 8 = 0 check** Ex 11-18**

Example 3 page 96Now 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 + 8½ b = 20

• THEN multiply both sides by the inverse of ½ . Which is 2!

2* ½ b = 2 * 20b = 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 m=3 right?• 2(3) –4 = 4(3) –10• 6 –4 = 12 –10• 2 = 2 YES!

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 railroad2(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 –8Combine the q’s on the right : 7q –6 = 8q –8Then subtract 7q from both sides7q –7q –6 = 8q – 7q – 8

-6 = q –8 Then add 8 to both sides -6 +8 = q –8 +8 give us2 = 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!– 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 102y/2 –1 = y/3 +1Looking 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)+63y-6 = 2y+63y-6+6=2y+6+63y=2y+123y-2y=2y-2y+12y=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=12.6The 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 = 126030p +804-804 =1260-804(1/30) 30p = (1/30) 456

p = 15.2Checking… 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)=13.4When x=60.5(6)+0.4(6+20)=?13.43+.4(26)=?13.43+10.4 =?13.413.4 = 13.4 Check!** Ex 35-38**

Some Quick Ones – Ex 4 page 104

a) b)2a-3=0 2k+5=3k+12a-3+3=0+3 2k-2k+5=3k-2k+12a =3 5=k+1(1/2)2a= (1/2)3 5-1=k+1-1a=3/2 or {3/2} 4=k or {4}And always check!

** Ex 39-54**

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 = xx+x = 2x 3x + 3 = 3(x+1)x+1 = x+1 yadda yadda5/x = 5/x (note x can’t equal 0)

Is it an identity? Example 5 pg 105

• Simplify both sides…by themselves7-5(x-6)+4=3-2(x-5)-3x+28Kill the ()’s 7-5x+30+4=3-2x+10-3x+28Group 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 106Solve 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 –2814 = -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…– 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 x+8– Five is added to a number x+5– Two more than a number x+2– A number is increased by 3 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– The product of 5 and a number 5x– Twice a number 2x– One half a number ½ x– Five percent a number 0.05x

Division

• Division– The ratio of a number to 6 x/6– The quotient of 5 and a number 5/x– Three divided by some number 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 12x and x-12 Check x – (x-12) = x-x+12 = 0 +12

• Two numbers with a sum of –8x and -8-x

Check x+ (-8-x) = x-x-8 = 0 –8 = -8The 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 3A 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 numbersa) 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 yy, 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(x-

3)• 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!– 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 ApplicationsExample 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 Problems1. 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 problemExample 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 feetTHEN 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 terms4.5x = 90 divide both sides by 4.5x= 90/4.5 = 20

So one angle is x = 20 degreesThe other is 3.5x = 3.5(20) = 70 degreesDO 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

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 ()’s2x+3x+105=255 simplify and subtract 105 from both sides5x=150 then divide by 5 x=30 Which is the answer 30mph.

** 23-26**

3(x+35)mi

3 hrx+35 mi/hr

Clear Road

2x mi2 hrx mi/hrIcy Road

DistanceTimeRate

D=RT

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

Since we know the total distance = going + coming55x=40(x+3) eliminate ()’s55x=40x+120 subtract 40x from both sides15x=120 divide both sides by 15so x=8 Going time = 8hours, Distance =55(8)=440miles

** 27-28**

40(x+3) mi

x+3 hr40 mphComing

55x mix hr55 mphGoing

DistanceTimeRate

D=RT

Work on Section 2.6Questions 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 pricex-0.12x=17,600 combine like terms (with x’s)0.88x=17,600 divide by 0.88x=17,600/0.88 x=$20,000 was the original price!

** 9-10**

Commission ProblemsExample 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 ProblemsExample 4 page 139

• Best to make your table!

CD Interest + mutual fund = total interest0.09x+0.10(2x) = 232 now we’ll solve it!

0.10(2x)2x10%Mutual Fund

0.09xx9%CD

Interest for 1 year

Amount Invested

Interest Rate

Ex 4 continued

0.09x+0.10(2x) = 2320.09x+0.20x=2320.29x =232x= 232/0.29x=$800 was invested in the CD2x=$1600 was invested in the Mutual Fund** 15-18**

Work on Section 2.7Questions 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 5b) –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=02(0) –3 <=? –50-3 <=? –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=13/36 <? 3(13/3) –5 <? 14 (the 3’s cancel)6 <? 13 –5 <? 146<? 9 <? 14 Yes! True.

** Ex 53-70**

Turning Words into ActionsExample 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 $500The washing machine costs x, the tax is 0.09xSo 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+4Looking up the perimeter equation from

somewhere 120<= 2W +2LSo 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 sidesx> 9-3x>6

b) -2x <= 6, x ___ -3divide 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