Download SS2.1 Simplifying Algebraic Expressions

SS2.1 Simplifying Algebraic Expressions
Term – Number, variable, product of a number and a variable or a variable
raised to a power.
Example: a) 5
b) 5x
c) xy
d) x2
Numeric Coefficient – The numeric portion of a term with a variable.
Example: What is the numeric coefficient?
a) 3x2
b) x/2
c) - 5x/2
d) – z
Like Term – Terms that have a variable, or combination of variables, that are
raised to the exact same power.
Example: Are the following like terms?
a) 7x
10x2
b) - 15z
23z
c) t
15t3
d) 5
5w
e) xy
6xy
2
f) x y
- 2x2y2
Simplifying an algebraic expression by combining like terms means adding
or subtracting terms in an algebraic expression that are alike. Remember
that a term in an algebraic expression is separated by an addition sign
(recall also that subtraction is addition of the opposite, so once you change
all subtraction to addition, you may simplify.) and if multiplication is
involved the distributive property must first be applied.
Step 1: Change all subtraction to addition
Step 2: Use the distributive property wherever necessary
Step 3: Group like terms
Step 4: Add numeric coefficients of like terms
Step 5: Don’t forget to separate each term in your simplified
expression by an addition symbol!!
Example: Simplify each
a) 2x + 4x2 + 5x  2x2 + 5
b) 5xy + 2 + 7  2xy
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c) 6x2y + 9 + 2xy
d) 6(x + y) + 2x + 5
e) 6x(2 + y) + y  xy
f) x2 + x(6  y)  xy
g) –(x + y) + 2y
h) 7  (x + 2)
Before we begin the next section, we should practice translation again, as
this chapter is building to word problems!
Example: Translate the following.
a) The quotient of 20 and 5 added to 9
b) Twice the sum of 11 and 7
c) The sum of two times 11 and 7
d) Three x added to the sum of twice x and one
e) Three x added to twice the sum of x and one
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SS 2.2 The Addition Property of Equality
Linear Equations in One Variable are equations that can be simplified to an
equation with one term involving a variable raised to the first power which is
added to a number and equivalent to a constant. Such an equation can be
written as follows:
ax + b = c
a, b & c are constants
a0
x is a variable
Our goal in this and the next section will be to rewrite a linear equation into
an equivalent equation (an equation with the same value) in order to arrive at
a solution. We will do this by forming the equivalent equation:
x=#
or
#=x
x is a variable
# is any constant
Remember that only 1 value will make a linear equation true and that is the
value that we desire. This also means that we can check our value and make
sure that the original equation, when evaluated at the constant, makes a true
statement.
Recall the Fundamental Theorem of Fractions:
a  c = a
(multiply/divide the numerator &
b  c
b
denominator by the same number and the resulting
fraction is equivalent)
The method for which we will be creating equivalent equations is very
similar to the Fundamental Theorem of Fractions.
Addition Property of Equality says that given an equation, if you add
(subtract) the same thing to both sides of the equal sign, then the new
equation is equivalent to the original.
a = b and a + c = b + c
are equivalent equations
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Example: Form an equivalent equation by adding the opposite of the
constant term on the left.
9 + x = 12
We’ll use the additive property of equality to move all variables to one side
of the equation (for simplicity, in the beginning, we will move variables to
the left) and all numbers to the other side (in the beginning to the right).
This is known as isolating the variable.
Isolating the variable is done by adding the opposite of the constant term to
both sides of the equation (the expression on each side of the equal sign
must first be simplified of course.) We add the opposite of the constant term
in order to yield zero! This makes the constant disappear from the side of
the equal sign with the variable! Don’t get to carried away and forget that
in order to make it disappear, you must add the opposite to both sides of the
equal sign!
Example: Solve by using the addition property of equality and the
additive inverse. In order to solve these, you must look at
the equation and ask yourself, “How will I make x stand
alone?” This is where the additive inverse comes in -You must add the opposite of the constant term in order to
get the variable to “stand alone.”
a) x + 9 = 11
b) x + 2 = 15
c) x  3 = 10
d) x  7/8 = 3/8
e) x + 3 = 0
Yes, you can probably look at each of the above equations and tell me what
the variable should be equivalent to, but the point here is to develop a
method that will work when the equation is more complicated!!
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How do we know that we got the right number for the above answers? We
check them using evaluation!
Example: Check to problems from the previous examplef
a) x = 2 so we replace x with 2 and get the following
2 + 9 = 11
11 = 11 true statement, therefore()this checks
b)
c)
d)
e)
Of course, every problem will not be as simplistic as the ones used as
examples so far, and sometimes we will have to simplify a problem before
we can solve it.
Example: Solve the following problems.
a) 6a  2  5a = -9 + 1
b) 5a + 6  4a = 7a + 8  7a
c) 3(a + 7/3)  2a = 1
d) - ½ (8a  12)  (-5a) = 4
e) 4a + 1  3a = 3(a + 2/3)  3a
Again we need to practice some translation problems. This time let’s
translate algebraic expressions into symbols, using some word and visual
problems.
Example: Write an algebraic equation to describe the picture and
solve.
Example: If I cut a rod that is 16 in. long into 2 pieces and one piece
is 7.2 in. long, how long is the other one? Write and solve
an algebraic equation.
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SS 2.3 The Multiplication Property of Equality and
SS 2.4 Solving Linear Equations
Mulitiplication Property of Equality says that two equations are equivalent if
the second is the same as the first, where both sides have been multiplied by
a nonzero constant.
a = b
is equivalent to ac = bc
We will be using this to solve such problems as:
Example: 4x = 24
 By inspection we see that if x = 6, both sides are equivalent!
 However, if we want to isolate x what must 4x be multiplied by? Think
about it’s reciprocal!
 Using the idea of isolating the variable and the multiplication property of
equality, we can arrive at a solution for x!
Example: Solve using the multiplicative property of equality
a) 16x = 48
b) x/3 = 36 (Recall that x/3 is equivalent to 1/3 x)
c) 14x  2 = 26
Now, let’s get a little more complicated with some material from section
four. We have all the tools that we need to solve any problem, we just have
to apply them in the correct order!
Example: Solve 12x  5 = 23  2x
Step 1: Move all numbers to one side(right)
Step 2: Move all variables to the other side(left)
Step 3: Isolate the variable(use multiplicative prop. of equality)
**Steps 1 & 2 are interchangeable. These steps both use the
additive property of equality.
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Example: Solve 2x + 10 + 14x = 2x  18
Step 1: Simplify each side of the equation
Step 2: Move all #’s to one side(right)
Step 3: Move all variables to the other side(left)
Step 4: Isolate x
Example: Solve 5(2x  1)  6x = 4
Step 1: Simplify each side of the equation(this includes
distributive property)
Step 2: Move all #’s to one side(right)
Step 3: Move all variables to the other side(left)
Step 4: Isolate x
Example: Solve 0.60(z  300) + 0.05z = 0.70z  0.41(500)
Step 1: This time simplifying may also include removing the
decimals, by multiplying both sides of the equation by
100. This means that every term gets multiplied by
100!!
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Example: Solve 4(5  w) = - w
3
Step 1: This time this may include clearing the equation of
fractions, by multiplying each term by the LCD!
Example: Solve x/2  1 = x/5 + 2
Step 1 should simplify the problem! You should,
1. clear fractions or decimals (if you prefer)
2. use the distributive property
3. combine all like terms on each side of the equation
Now, here are some more to practice in class. Raise your hand if you
especially need help. We will do these together in a few minutes, but you
are expected to try them on your own, while I circulate.
Your Turn
Example: Solve
a) -7  (10x + 3) = 5x  (x  18)
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b) ½ (x  2) = 5
c) 2(a + 7) = 5a  4
3
d) 0.5(0.25 + x) = 0.75(x  0.5)
Now, for a couple of cases that might trip you up if you are not aware that
such a solution can exist!
Example: (The no real solution case)
2x  1 = 1 + 2x
Example: (The all real number solution case)
7(x + 1) + 2 = 7x + 9
Now, for some more translation problems:
Example: Write an equation and solve.
a) Five more than twice a number is 37. What is the number?
b) Gary and Sue are 25 years old, if you sum their ages. Gary
is one year older than Sue. How old are Gary and Sue?
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SS 2.5 An Introduction to Problem Solving
At this point we are about to begin the sections on word problems. I would
like to give you a list of words and phrases that appear in word problems
coupled with how they can be translated.
Words and Phrasing for Word Problems, by Operation
Note: Let any unknown be the variable x.
Addition
Word
sum
more than
added to
greater than
increased by
years older
than
Phrasing
The sum of a number and 2
5 more than some number
Some number added to 10
7 greater than some number
Some number increased by 20
15 years older than John
Algebraic Expression
x + 2
x + 5
10 + x
x + 7
x + 20
x + 15
Note: Because addition is commutative, each expression can be written
equivalently in reverse, i.e. x + 2 = 2 + x
Subtraction
Word
difference of
years
younger than
diminished
by
less than
decreased by
subtract from
Phrasing
The difference of some number and 2
The difference of 2 and some number
Sam's age if he is 3 years younger
than John
15 diminished by some number
Some number diminished by 15
17 less than some number
Some number less than 17
Some number decreased by 15
15 decreased by some number
Subtract some number from 51
Subtract 51 from some number
Multiplication
Word
Phrasing
product
The product of 6 and some number
times
24 times some number
twice
Twice some number
Twice 24
multiplied by 8 multiplied by some number
at
Some number of items at $5 a piece
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Algebraic Expression
x  2
2  x
x  3
15  x
x  15
x  17
17  x
x  15
15  x
51  x
x  51
Algebraic Expression
6x
24x
2x
2(24)
8x
$5x
"fractional
part" of
"Amount" of
"$" or "¢"
percent of
x
A quarter of some number
¼ x or
/4 .
Amount of money in some number of
dimes (nickels, quarters, pennies, etc.)
3 percent of some number
0.1x (dollars) or 10x (cents)
0.03x
Note: Because multiplication is commutative all of the above algebraic
expressions can be written equivalently in reverse, i.e. 6x = x6
Division
Word
quotient
divided by
ratio of
Phrasing
The quotient of 6 and some number
The quotient of some number and 6
Some number divided by 20
20 divided by some number
The ratio of some number to 8
The ratio of 8 to some number
Algebraic Expression
6  x
x  6
x  20
20  x
x  8
8  x
Note: Division can also be written in the following equivalent ways, i.e. x 
6 = x/6 = 6x = x
6
Exponents
Words
squared
square of
cubed
cube of
(raised) to
the power of
Phrasing
Some number squared
The square of some number
Some number cubed
The cube of some number
Some number (raised) to the power of
6
Algebraic Expression
x2
x2
x3
x3
x6
Equals
Words
Phrasing
Algebraic Equation
is
The sum of 5 and 4 is 9.
5 + 4 = 9
will be
12 decreased by 4 will be 8.
12  4 = 8
was
The quotient of 12 and 6 was 2.
12  6 = 2
Note: Any form of the word is can be used to mean equal.
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Percentages
Rewrite all problems into this form
------% of -----(whole) is --------(part)
Make Algebraic Equation
Percent Missing
72x = 36
Note: x will be a decimal, convert to %
Whole Missing
0.5x = 36
Note: Change percent to a decimal to
solve
Part Missing
0.5(72) = x
Note: Change percent to a decimal to
solve
Note: If the whole is greater than the part, then the percentage will be greater than or equal to 100%
Steps to Solving a Word Problem
1. Understand what the problem says and what is being asked of you.
2. Assign a variable to an unknown and put all other unknowns in terms of
this one variable.
3. Draw a picture or make a table to help you with the problem.
4. Translate the problem into an equation.
5. Solve the equation that you have made for yourself.
6. Check the solution and answer the question in a complete sentence or
phrase with a label.
Examples
1) The larger of two numbers is five more than three times the smaller
number. If their sum is 29, find the numbers.
2) The age of Boris is seven years more than twice the age of Ivan. If the
sum of their ages is 52, find the age of Boris.
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3) Marta recently paid $290 for a bicycle and a helmet. She paid two
dollars more than five times as much for the bike as the helmet. What was
the price of the bike? (Note: The phrase “as much as” can be tricky because
sometimes it is split up. The as much as is what is being multiplied and the
phrasing “for such and such” is what the product is equivalent to. ie 5helmet
= bike)
4) Abdul bought two books and three computer disks. Each book cost twice
as much as each disk (including tax). If his total bill was exactly $21, how
much did he pay for each book?
5) In a certain class, there are eight more than twice as many women as
men. If the class has 47 students, how many women are in the class?
6) Find two numbers such that the larger is five less than three times the
smaller. When the larger is multiplied by four and then decreased by three
times the smaller the result is 25.
7) The weight of Tarzan is 30 pounds less than twice the weight of
Madonna. Four times Madonna's weight added to three times Tarzan's
weight is 960 pounds. Find the weight of Ferd Burfil who weighs five
pounds less than Tarzan.
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8) Gilda is seven years less than four times the age of Ivan. When Ivan's
age is multiplied by five and increased by twice the age of Gilda, the result
is 64. How old is Gilda?
9) The larger of two numbers is five less than three times the smaller.
Seven times the larger diminished by twice the smaller is 41. Find the larger
number.
10) For her exercise workout, Olga rode her bike a distance of three miles
more than twice the distance that she jogged. If the total distance she
traveled was 21 miles, how far did she ride her bike?
11) Mortimer is three years older than twice the age of Alfonso. If the sum
of their ages is 51, find the age of Mortimer.
12) The cost of the speakers for Corine's new stereo system was $45 more
than three times the cost of the receiver. What did her speakers cost if the
total system cost $929?
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13) Carol recently bought a book, a computer disk, and a pogo stick, paying
a total of $126. She paid two dollars more than three times as much for the
book as the disk, while the pogo stick cost twice as much as the book. Find
the cost of the pogo stick.
14) Connie used her new $450 bike to travel the six miles to the math party.
During the second hour of the party she worked eight more than twice as
many problems as Kim, but their combined total of 47 problems was the
highest pair at the party. If the party ended at 9PM sharp, how many
problems did Connie work?
15) Ivan slept two hours less than Carlos, while Boris slept as much as Ivan
and Carlos combined . If these guys slept a total of 32 hours, how many
hours did Boris sleep?
16) The larger of two numbers is six less than twice the smaller. When five
times the smaller is increased by three times the larger, the result is 114.
Find the sum of these two numbers.
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Supplemental Material – Ratios and Proportions
Ratios and Proportions
A ratio is a quotient of two numbers where the divisor isn't zero. A ratio is
stated as: a to b
a : b or a where a & b are whole numbers and b0
b
A proportion is a mathematical statement that two ratios are equal.
2 = 4
3
6
is a proportion
It is read as: 2 is to 3 as 4 is to 6
The numbers on the diagonal from left to right, 2 and 6, are called the
extremes
The numbers on the diagonal from right to left, 4 and 3, are called the
means
If we have a true proportion, then the product of the means equals the
product of the extremes.
These are also called the cross products and finding the product of the means
and extremes is called cross multiplying.
Example:
Find the cross products of the following to show that this
is a true proportion
27 = 3
72
8
The idea that in a true proportion, the cross products are equal is used to
solve for unknowns!
Example:
a = 12
25
10
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Example:
x + 1 = 2
2x + 3
3
Truly, however, the most useful thing about ratios and proportions are their
usefulness in word problems.
The key is to set up equal ratios of one thing to another
Example: The ratio of the weight of an object on Earth to the weight
of the same object on Pluto is 100 to 3. If an elephant
weighs 4100 pounds on Earth, find the elephant’s weight
on Pluto.
1) Set up words
2) Fill in with numbers
3) Solve for missing
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Example: There are 110 calories per 28 2/5 grams of Crispy Rice
cereal. find how many calories are in 42 3/5 grams of this
cereal.
1) Set up words
2) Fill in with numbers
3) Solve for missing
Example: Miss Rocky’s new Miata gets 35 miles per gallon. Find
how far she can drive if the tank contains 13.5 gallons of
gas.
1) Set up words
2) Fill in with numbers
3) Solve for missing
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Example: If Sam Abney can travel 343 miles in 7 hours, find how far
he can travel if he maintains the same speed for 5 hours.
1) Set up words
2) Fill in with numbers
3) Solve for missing
Example: Mr. Lin’s contract states that he will be paid $153 per 8
hour day to teach mathematics. Find how much he earns
per hour, rounded to the nearest cent.
**You can do the problems that I stated to do in SS6.7 now, except problem
number 34.
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SS 2.6 Formulas and Problem Solving
Sometimes word problems will involve a relationship that already has a
known formula. In this case it is our job to figure out the known formula
being asked of us, the quantities that are given and solve the problem for the
unknown quantity.
Once we identify the formula needed to solve a problem, then we need only
evaluate and solve the formula with the information given in the problem at
hand.
Some common formulas are:
Area of a Rectangle
A=l*w
Simple Interest
I = PRT
where l = length and w = width
where P = Principle , R = Rate
and T = Time
Perimeter of Rectangle
P = 2l + 2w
where l = length and w = width
Distance
d=r*t
where r = rate and t = time
Volume of a Rectangular Solid
V=l*w*h
where l = length , w = width and h = height
Degrees Fahrenheit
F = 9/5 C + 32
where C = Degrees Celsius
Perimeter of a Triangle
P = S1 + S2 + S3 where S1 = Side 1 , S2 = Side 2
and S3 = Side 3
Examples
1) A triangle has a perimeter of 78 meters. Two sides have the same length
and the third side is six meters less than the sum of the other two sides. Find
the length of the largest side.
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2) The longest side of a triangle is 4 cm more than three times the shortest
side. The third side is twice the shortest side. If the perimeter is 46 cm, find
the length of the largest side.
3) The floor of Batman's costume storage closet is nine feet more than three
times its width. If the perimeter is 138 feet, what is its length?
4) Fannie Farkel selected a rectangular frame to place the picture of Ferd
Burfil. The length of the frame was 8 cm less than twice its width. Find the
dimensions of the frame if its perimeter was 68 cm.
5) Jim put $5000 in the bank for 2 years at 8 percent simple interest. How
much money did he get when he withdrew his money after 2 years?
6) Maria deposited $800 at 8 percent simple interest. At the end of 3 years
she will have how much money?
7) Helen could travel 40 miles per hours. How long would it take her to
travel 600 miles?
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8) Mildred traveled the first 120 miles in 3 hours. What was her speed?
How long would it take her to travel 400 miles at the same speed?
9) If it is 52 F, what is the temperature to the nearest degree in Celsius?
10) If it is 27 Celsius, to the nearest degree what is the temperature in
Fahrenheit?
11) Cereal A comes in a rectangular box that is 20 centimeters wide, 6
centimeters deep and 25 centimeters high. What is the volume of the cereal
box?
12) What is the length of a fish tank, which is twice as high as it is wide and
1.5 feet wide if its volume is 9 cubic feet?
13) Find the area of the bottom piece of glass on the above fish tank.
* Problems are from Prealgebra: Thinking Like a Mathematician , Jim Symons, Second Edition, McGraw
Hill, Inc., 1995. Algebra ½: An Incremental Development, Saxon, John Jr., Second Edition, Saxon
Publishing Co., 1997. and SRA Spectrum Math: Purple, Fourth Edition, McGraw Hill, Inc., 1997.
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SS 2.7 Percent and Problem Solving
Recall that a percent is a part of a hundred.
We can write percentages as parts of 100, but since a percentage is a fraction
in fractional form it needs to be reduced.
Example: 75% = .75 = 75 = 3
100
4
All these are called equivalent forms, because they indicate the same thing
Example: Express 25 % as a reduced fraction
Example: Express 2/7 as a percentage
Example: Express .982 as a percentage
Here is a shortcut for converting a decimal to a percentage and a percentage
to a decimal:
Decimal to a Percentage  Move the decimal two places to the
right
Percentage to a Decimal  Move the decimal two places to the left
We will be seeing percentage problems in word problems. The following
will be the forms that we will be seeing and their corresponding algebraic
equation.
Forms
Percent of a number is what
What percent of a number is the total
Percent of what is the total
Algebra Equation
(%)(number) = x
(x%)(number) = total
(%)(x) = total
The key to remember is that: % of something means multiplication
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Example: 35% of 18 is what?
Example: 16% of 10 is what?
Example: _____% of 10 is 6?
Example: 17% of _______ is 12?
Example: 81% of _______ is 62?
The key to doing word problems with percentages is to remember the form
and to put each problem in that form! % * whole = part
Example: I am 20% shorter than my dad. My dad is 75 inches tall.
How tall am I?
Example: Tax is 7 ¼ % in some areas of California. What amount of
tax will be paid on an item that costs $12.97?
Example: The price on the item was $7.25, but I had to pay $7.85 for
the item. What was the tax?
Example: Bonnie and Clyde rob banks for a living. In each robbery,
Bonnie spends 7 hours planning and 2 hours in the actual
robbery. Clyde on the other hand spends 10 hours
planning and 2 hours in the robbery. What percent of the
total time does Bonnie help? If they rob a band and get
$17,761, how much should Bonnie get? Clyde?
Example: Tom and Huck spent 18% of the time that they told Tom's
aunt it took them to white was the fence fishing. If they
spent 7 hours fishing, how long, to the nearest hour, did
they tell Aunty that it took them to paint the fence?
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Precent increase or decrease problems always involve the original price.
They are usually 2 step problems, where you must first find the amount of
increase or decrease, and then find the percentage of the original (old)
price.
% of old = increase/decrease
Example: Hallahan’s Construction Company increased their estimate
for building a new house from $95,500 to $110,000. Find
the percent increase.
SS 2.8 Further Problem Solving
This section is a continuation of word problems. I will do two examples of
each type of problem in this section, but we will not spend extensive time on
problems, since each type is worked the same way, and it takes practice on
your part.
Type 1: Geometry Problems
Important Keys: Area Formulas
Square/Rectangle
Triangle
Perimeter Formulas
Square/Rectangle
Triangle
Definitions of geometric figures
Isosceles Triangle
Equilateral Triangle
Example: An architect designs a rectangular flower garden such
that the width is exactly two-thirds of the length. If 260
feet of antique picket fencing are to be used, find the
dimensions of the garden.
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Example:
A square animal pen and a pen shaped like an equilateral
triangle have equal perimeters. Find the length of the
sides of each pen if the sides of the triangular pen are
fifteen less than twice a side of the square pen.
Type 2: Distance Problems
Important keys to remember for this type of problem are the formula
D=R*T
Example:
A jet plane traveling at 500 mph overtakes a propeller
plane traveling at 200 mph that had a 2 hour head start.
How far from the starting point are the planes?
Example:
Two hikers are 11 miles apart and walking toward each
other. They meet in 2 hours. Find the rate of each hiker
if one hiker walds 1.1 miles per hour faster than the
other.
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Type 3: Mixture Problems
Keys to solving are remembering that these are percentage problems and
building a table like the following!
Amount Solution *
Strength =
Amount of Pure
Example:
How many cubic centimeters of a 25% anitbiotic solution
should be added to a 10 cubic centimeters of a 60%
antibiotic solution in order to get a 30% antibiotic
solution?
Example:
How much water should be added to 30 gallons of a
solution that is 70% antifreeze in order to get a mixture
that is 60% antifreeze?
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Type 4: Interest Problems
Keys to solving interest problems are to build a chart as the following and
remember that P*R*T = I
Principle
*
Rate
*
Time
*
Interest
Example:
I have $2000 dollars to invest. I wish to invest it in such
a way that I will earn $188 dollars from two accounts.
One account pays 8% interest and the other 10%
annually. How much should I invest in each account?
Example:
Suppose that you invest a certain amount of money in an
account that earns 8% in annual interest. At the same
time, you invest $2,000 more that that in an account that
pays 9% in annual interest. If the total interest in both
accounts at the end of the year is $690, how much is
invested in each account?
Note: Mixture and interest problems are very similar in form!
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SS 2.9 Linear Inequalities
Review of Inequality Symbols




Graphing of simple inequalities (contain only one inequality)
 -- Graph with an open circle and a line to the left, if in standard
form
 -- Graph with an open circle and a line to the right, if in standard
form
 -- Graph with a closed circle and line to the right, if in standard
form
 -- Graph with a closed circle and a line to the left, if in standard
form
Examples: Graph each of the following (on a number line)
a) x  3
b) 2  y
c) -2  v
d) t  5
Parts b and c in the example are examples of inequalities written in
nonstandard form. To read an inequality written in nonstandard form, we
must read right to left, instead of our usual left to right. Thus it is easiest to
always put inequalities in standard form before reading and trying to graph
them.
Examples: Put the following in standard form
a) 5  x
b) –2  y
c) 5002  m
When putting an inequality into standard form, pay attention to what the
arrow is pointing at and make it the same when written in standard form.
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Graphing of compound inequalities (contains 2 inequalities):
A compound inequality is read as an “and”
Example:
15  x  20 is read as:
x is greater than 15 and x is less than 20
Example: Graph the following compound inequalities
a) 8  x  9
b) –2  x  5
c) 5  x  -2
d) 5  x  -2
Solving Linear Inequalities
If all we must do to solve a linear inequality is to add or subtract then
they are solved in the exact same manner as a linear equation. Recall the
steps to solving a linear equation covered in SS2.3 & SS2.4.
Example: Solve and graph the linear inequality
a)
x + 2  7
b)
2x + 7  9 + x
c)
2x  15 + x  15 + 2(x  1)
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If we must multiply or divide by a positive number to get the solution, then
the steps for finding the solution are still the same!
Example: Solve and graph the linear inequality
5x  15 + 2  15 + 3x
The problem occurs when I must multiply or divide by a negative number!
Whenever I multiply or divide by a negative number, the sense of the
inequality must reverse.
Example: Solve and graph
a) 3/4x  1  2x + ¼
b) -2x + 5  9
To see why it is true that the sense of the inequality must reverse, let’s
rework these problems and keep the numeric coefficient of the variable
positive. Notice that you must read the inequality in nonstandard form
(right to left) and the answer is the same as above.
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Solving Compound Linear Inequalities
Step 1: See the problem as having 3 parts – a left, a middle and a
right.
Step 2: Simplify the three parts if necessary
Step 3: Remove all constants from the middle by adding their
opposite to all three part, the left and right sides of the
inequality and the middle
Step 4: Remove the numeric coefficient of the variable term (middle)
by multiplying each part by the reciprocal (don’t forget to
reverse the sense of both inequalities if your numeric
coeffiecient is negative!)
Step 5: Check your answer
Example: Solve and graph each compound inequality
a) 2  x  1  5
b) 2  3/4x  1  2 ¼
Don’t forget to switch the sense of the inequality if you must multiply or
divide by a negative number!
Example: Solve and graph
7  -(x + 5)  2
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