SS2.1 Simplifying Algebraic Expressions

Term – Number, variable, product of a number and a variable or a variable raised to a power.

Example: a) 5b) 5xc) xyd) 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) – zLike 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 23zc) t 15t3

d) 5 5we) xy 6xyf) x2y - 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 additionStep 2: Use the distributive property wherever necessaryStep 3: Group like termsStep 4: Add numeric coefficients of like termsStep 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 = ca, b & c are constantsa 0x 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 + care 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!!

How do we know that we got the right number for the above answers? We check them using evaluation!

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Example: Check to problems from the previous examplefa) x = 2 so we replace x with 2 and get the following2 + 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.

SS 2.3 The Multiplication Property of Equality and SS 2.4 Solving Linear Equations

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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 equalitya) 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 2xStep 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.

Example: Solve 2x + 10 + 14x = 2x 18Step 1: Simplify each side of the equationStep 2: Move all #’s to one side(right)

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Step 3: Move all variables to the other side(left)Step 4: Isolate x

Example: Solve 5(2x 1) 6x = 4Step 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!!

Example: Solve 4(5 w) = - w 3

Step 1: This time this may include clearing the equation of

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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 property3. 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 TurnExample: Solve

a) -7 (10x + 3) = 5x (x 18)

b) ½ (x 2) = 5

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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?

SS 2.5 An Introduction to Problem SolvingAt 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.

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Words and Phrasing for Word Problems, by OperationNote: Let any unknown be the variable x.

AdditionWord Phrasing Algebraic Expression

sum The sum of a number and 2 x + 2more than 5 more than some number x + 5added to Some number added to 10 10 + xgreater than 7 greater than some number x + 7increased by Some number increased by 20 x + 20years older than

15 years older than John x + 15

Note: Because addition is commutative, each expression can be written equivalently in reverse, i.e. x + 2 = 2 + x

SubtractionWord Phrasing Algebraic Expression

difference of The difference of some number and 2The difference of 2 and some number

x 22 x

years younger than

Sam's age if he is 3 years younger than John

x 3

diminished by

15 diminished by some numberSome number diminished by 15

15 xx 15

less than 17 less than some numberSome number less than 17

x 1717 x

decreased by Some number decreased by 1515 decreased by some number

x 1515 x

subtract from Subtract some number from 51Subtract 51 from some number

51 xx 51

MultiplicationWord Phrasing Algebraic Expression

product The product of 6 and some number 6xtimes 24 times some number 24xtwice Twice some number

Twice 242x2(24)

multiplied by 8 multiplied by some number 8xat Some number of items at $5 a piece $5x"fractional part" of

A quarter of some number ¼ x or x/4 .

"Amount" of "$" or "¢"

Amount of money in some number of dimes (nickels, quarters, pennies, etc.)

0.1x (dollars) or 10x (cents)

percent of 3 percent of some number 0.03x

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Note: Because multiplication is commutative all of the above algebraic expressions can be written equivalently in reverse, i.e. 6x = x6

DivisionWord Phrasing Algebraic Expression

quotient The quotient of 6 and some numberThe quotient of some number and 6

6 xx 6

divided by Some number divided by 2020 divided by some number

x 2020 x

ratio of The ratio of some number to 8The ratio of 8 to some number

x 88 x

Note: Division can also be written in the following equivalent ways, i.e. x 6 = x/6 = 6x = x

6Exponents

Words Phrasing Algebraic Expressionsquared Some number squared x2

square of The square of some number x2

cubed Some number cubed x3

cube of The cube of some number x3

(raised) to the power of

Some number (raised) to the power of 6

x6

EqualsWords Phrasing Algebraic Equation

is The sum of 5 and 4 is 9. 5 + 4 = 9will be 12 decreased by 4 will be 8. 12 4 = 8was The quotient of 12 and 6 was 2. 12 6 = 2Note: Any form of the word is can be used to mean equal.

PercentagesRewrite all problems into this form Make Algebraic Equation

------% of -----(whole) is --------(part) Percent Missing 72x = 36Note: x will be a decimal, convert to %Whole Missing 0.5x = 36

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Note: Change percent to a decimal to solvePart Missing 0.5(72) = xNote: 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.

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

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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.

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?

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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?

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.

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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.

Supplemental Material – Ratios and Proportions

Ratios and Proportions

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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 b0 b

A proportion is a mathematical statement that two ratios are equal.

2 = 4 is a proportion 3 6

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

Example: x + 1 = 2 2x + 3 3

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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

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

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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

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

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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.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

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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 RectangleA = l * w where l = length and w = width

Simple InterestI = PRT where P = Principle , R = Rate

and T = TimePerimeter of Rectangle

P = 2l + 2w where l = length and w = width

Distanced = r * t where r = rate and t = time

Volume of a Rectangular SolidV = l * w * h where l = length , w = width and h = height

Degrees FahrenheitF = 9/5 C + 32 where C = Degrees Celsius

Perimeter of a TriangleP = S1 + S2 + S3 where S1 = Side 1 , S2 = Side 2

and S3 = Side 3Examples

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.

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.

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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?

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?

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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.

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.

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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 Algebra EquationPercent of a number is what (%)(number) = xWhat percent of a number is the total (x%)(number) = totalPercent of what is the total (%)(x) = total

The key to remember is that: % of something means multiplication

Example: 35% of 18 is what?

Example: 16% of 10 is what?

Example: _____% of 10 is 6?

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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?

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.

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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 ProblemsImportant Keys: Area Formulas

Square/RectangleTriangle

Perimeter FormulasSquare/RectangleTriangle

Definitions of geometric figuresIsosceles TriangleEquilateral 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.

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.

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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.

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

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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?

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

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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!SS 2.9 Linear Inequalities

Review of Inequality Symbols

Graphing of simple inequalities (contain only one inequality)

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-- 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 5Parts 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 forma) 5 xb) –2 yc) 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.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 inequalitiesa) 8 x 9

b) –2 x 5

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c) 5 x -2

d) 5 x -2

Solving Linear InequalitiesIf 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 inequalitya) x + 2 7

b) 2x + 7 9 + x

c) 2x 15 + x 15 + 2(x 1)

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 inequality5x 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.

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Example: Solve and grapha) 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.

Solving Compound Linear InequalitiesStep 1: See the problem as having 3 parts – a left, a middle and a

right.Step 2: Simplify the three parts if necessaryStep 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

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Example: Solve and graph each compound inequalitya) 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 graph7 -(x + 5) 2

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