MATH 010

JIM DAWSONJIM DAWSON

1.1 INTRODUCTION TO INTEGERS

This section is an introduction to:

• Positive Integers

• Negative Integers

• Opposites

• Additive Inverse

• Absolute Value

1.2 ADDING AND SUBTRACTING INTEGERS If the signs are the same : ADD the absolute If the signs are the same : ADD the absolute

values and place the common sign in the values and place the common sign in the answer.answer.

6+7=136+7=13

- 13+(-5)= -18

If the signs are different : SUBTRACT the If the signs are different : SUBTRACT the absolute value of the smaller number from absolute value of the smaller number from the absolute value of the larger number. the absolute value of the larger number. Place the sign of the larger number using Place the sign of the larger number using absolute value in the answer.absolute value in the answer.

14+(-6)= 8

-21+10= -11

1.3 MULTIPICATION AND DIVISION OF INTEGERS1.1. Determine the sign of the answer first:Determine the sign of the answer first:

Count the negative signs: even number of negative signs – the answer is positive

An odd number of negative signs- the answer is negative

2.Multiply or Divide using the absolute values2.Multiply or Divide using the absolute values

-5 x (-6)=30

7 x (-4)= -28

1.4 REVIEW OF FRACTIONS AND DECIMALS WITH SIGNS The rules for sign are the same for The rules for sign are the same for

fractions and decimals as they were for fractions and decimals as they were for integers.integers.

CONVERTING BETWEEN FRACTIONS, DECIMALS, AND PERCENTS Change a percent to a decimal.Change a percent to a decimal.

Move the decimal point TWO Move the decimal point TWO places from right to left.places from right to left.

Change a decimal to percent.Change a decimal to percent.Move the decimal point TWO Move the decimal point TWO

places from left to right.places from left to right.

FRACTION TO A PERCENT

Change the fraction to a Change the fraction to a decimal(numerator divided by decimal(numerator divided by denominator) and move the denominator) and move the decimal point TWO places from decimal point TWO places from left to right.left to right.

CHANGE A PERCENT WITH A FRACTION TO A FRACTION

Drop the % and multiply by 1 over 100.

•EXPONENTIAL NOTATION AND SOLVING EXPONENTS If the base is negative and does not have If the base is negative and does not have

parentheses around it the sign of the answer parentheses around it the sign of the answer is ALWAYS negative.is ALWAYS negative.

If the base is negative and has parentheses If the base is negative and has parentheses around it; look at the exponent to find the around it; look at the exponent to find the sign of the answersign of the answer

Even numbered exponent: positive answerEven numbered exponent: positive answer Odd numbered exponent: negative answer Odd numbered exponent: negative answer

ORDER OF OPERATIONS AGREEMENT1.1. Priority #1-GROUPING SYMBOLSPriority #1-GROUPING SYMBOLS

2.2. Priority #2- EXPONENTSPriority #2- EXPONENTS

3.3. Priority #3- MULTIPLY AND DIVIDE Priority #3- MULTIPLY AND DIVIDE AS THEY OCCUR FROM LEFT TO AS THEY OCCUR FROM LEFT TO RIGHTRIGHT

4.4. Priority #4- ADD AND SUBTRACT AS Priority #4- ADD AND SUBTRACT AS THEY OCCUR FROM LEFT TO RIGHTTHEY OCCUR FROM LEFT TO RIGHT

TRANSLATE AND SIMPLIFY

The translation must be done first and then The translation must be done first and then simplify using the rules learned previously simplify using the rules learned previously in the chapter.in the chapter.

The answer must be in descending order.The answer must be in descending order.

2.1 EVALUATING VARIABLE EXPRESSIONS

COMBINING LIKE TERMSCOMBINING LIKE TERMS Add or subtract the terms with the same Add or subtract the terms with the same

variable partvariable partPlace the answer in descending orderPlace the answer in descending order2a+3b-4a+7b=2a-4a+3b+7b2a+3b-4a+7b=2a-4a+3b+7b-2a+10b-2a+10b

2.2 SIMPLIFYING VARIABLE EXPRESSIONS Combining like termsCombining like terms

Combine the terms with the same Combine the terms with the same variable part or the constantsvariable part or the constants-3a+7+5a-9=-3a+5a and7-9-3a+7+5a-9=-3a+5a and7-92a-2 2a-2

MULTIPLYING VARIABLE TERMS Multiply the number parts and bring the Multiply the number parts and bring the

variable into the answer.variable into the answer. -3x(5)=-3(5)=-15x-3x(5)=-3(5)=-15x 7(-4a)=-7(4)=-28a7(-4a)=-7(4)=-28a (-2b)(-6)=-2(-6)=12b(-2b)(-6)=-2(-6)=12b

MULTIPLYING VARIABLE TERMS Multiply the number parts and bring the Multiply the number parts and bring the

variable into the answer.variable into the answer. -6(-4a)=-6(-4)= 24a-6(-4a)=-6(-4)= 24a (-5x)(-3)=(-5)(-3)=15x (-5x)(-3)=(-5)(-3)=15x

APPLYING THE DISTRIBUTIVE PROPERTY The Distributive Property is used to remove The Distributive Property is used to remove

parentheses.parentheses. If the terms inside the parentheses are If the terms inside the parentheses are

different, multiply the term on the outside different, multiply the term on the outside by every term on the inside.by every term on the inside.

Place the answer in descending order.Place the answer in descending order. 3(2x-4)=3(2x) and 3(-4)=6x-123(2x-4)=3(2x) and 3(-4)=6x-12

If you cannot combine like terms inside the If you cannot combine like terms inside the parentheses, multiply the outside term by parentheses, multiply the outside term by each term inside the parentheses.each term inside the parentheses. -3(4x+2)=-3(4x) and –3(2)-3(4x+2)=-3(4x) and –3(2) -12x-6-12x-6 Place the answer in descending orderPlace the answer in descending order

SIMPLIFYING A GENERAL VARIABLE EXPRESSION Use the Distributive Property to remove Use the Distributive Property to remove

parentheses and bracketsparentheses and brackets Combine like terms when possibleCombine like terms when possible Place the answer in descending orderPlace the answer in descending order Be careful with sign!Be careful with sign!

2.3 TRANSLATING VERBAL EXPRESSIONS Memorize the expressions on p.67Memorize the expressions on p.67 Rules for ParenthesesRules for Parentheses

Use parentheses to infer multiplication Use parentheses to infer multiplication when neededwhen needed

Use parentheses to separate two Use parentheses to separate two processes together not separated by a processes together not separated by a number or a variablenumber or a variable

Use parentheses to separate a more than Use parentheses to separate a more than or less than phrase with a number and or less than phrase with a number and letter next to the phrase from any other letter next to the phrase from any other phrase in the expressionphrase in the expression

EXAMPLES OF TRANSLATING VERBAL EXPRESSIONS

7 ADDED TO 3 LESS THAN A NUMBER7 ADDED TO 3 LESS THAN A NUMBER 7+(n-3)7+(n-3)

4 TIMES THE DIFFERENCE BETWEEN 4 TIMES THE DIFFERENCE BETWEEN A NUMBER AND 4A NUMBER AND 4 4(n-4)4(n-4)

THE SUM OF 2 AND THE PRODUCT OF THE SUM OF 2 AND THE PRODUCT OF A NUMBER AND 9A NUMBER AND 9

EXAMPLES OF TRANSLATING

2+9x2+9x 6 TIMES THE TOTAL OF A NUMBER 6 TIMES THE TOTAL OF A NUMBER

AND 8AND 8 6(n+8)6(n+8)

5 INCREASED BY THE DIFFERENCE 5 INCREASED BY THE DIFFERENCE BETWEEN 10 TIMES a AND THREEBETWEEN 10 TIMES a AND THREE 5+(10a-3)5+(10a-3)

TRANSLATE AND SIMPLIFY

Translate the verbal expression FIRST and Translate the verbal expression FIRST and then simplify using the rules that were then simplify using the rules that were applied earlier in the chapter.applied earlier in the chapter.

The answer must be in descending order if The answer must be in descending order if the expression was able to be simplified.the expression was able to be simplified.

DEFINING THE UNKNOWNS

In order to define an unknown quantity, In order to define an unknown quantity, assign a variable to that quantity, and then assign a variable to that quantity, and then attempt to express other unknown quantities attempt to express other unknown quantities in terms of the same variable.in terms of the same variable.

These are equations of one variable; These are equations of one variable; therefore, the same variable must be used therefore, the same variable must be used when defining any unknowns.when defining any unknowns.

3.1 SOLVING EQUATIONS OF THE FORM x+a=b The Addition Property of EquationsThe Addition Property of Equations

The goal is to solve for the unknownThe goal is to solve for the unknownVARIABLE = CONSTANTVARIABLE = CONSTANT

Find the number that is on the same side of Find the number that is on the same side of the equation as the variable and use the the equation as the variable and use the opposite process on both sides of the opposite process on both sides of the equation to solve the unknown quantity. equation to solve the unknown quantity.

SOLVING EQUATIONS USING THE ADDITION PROPERTY x+a=b X+4=12X+4=12

Find the number that is on the same side of the Find the number that is on the same side of the equation as x and use the opposite process to equation as x and use the opposite process to remove the number from the x side . The remove the number from the x side . The number to be removed is 4. It`s opposite is –4. number to be removed is 4. It`s opposite is –4. The unknown can be solved by the following; The unknown can be solved by the following; x+4-4=12-4;x=8x+4-4=12-4;x=8

You must do the same thing on both sides of You must do the same thing on both sides of the equation; -4 on both sidesthe equation; -4 on both sides

SOLVE AN EQUATION OF THE FORM ax=b Use the Multiplication Property of Equations to Use the Multiplication Property of Equations to

solve the unknownsolve the unknown Find the number that is on the same side of the Find the number that is on the same side of the

equation as the unknown and multiply both equation as the unknown and multiply both sides by the reciprocal and the sign that is with sides by the reciprocal and the sign that is with the number.the number.

Apply the Division Principle as a shortcut with Apply the Division Principle as a shortcut with integers and decimals when possible. integers and decimals when possible.

THE BASIC PERCENT EQUATION Percent x Base = AmountPercent x Base = Amount

P x B =AP x B =A 20% of what number is 30?20% of what number is 30? Translate and solve the verbal Translate and solve the verbal

expression. Change the percent to a expression. Change the percent to a decimal or fraction.decimal or fraction.0.20 x n = 30; Solve for n; 30 divided 0.20 x n = 30; Solve for n; 30 divided

0.20 = n; n = 150 0.20 = n; n = 150

MORE EXAMPLES OF THE BASIC PERCENT EQUATION 70 is what percent of 80?70 is what percent of 80? Translate and solve. Change the answer to a Translate and solve. Change the answer to a

percent.percent. 70 = n x 80; 70 divided by 80 = 0.875 which is 70 = n x 80; 70 divided by 80 = 0.875 which is

87.5%; n = 87.5%87.5%; n = 87.5% What is 40% of 80?What is 40% of 80? Translate and solve. Change the percent to a Translate and solve. Change the percent to a

decimal. decimal. 40% = 0.40;n = 0.40 x 80;n =3240% = 0.40;n = 0.40 x 80;n =32

3.2 GENERAL EQUATION PART 1 Solve an equation of the form ax+b=cSolve an equation of the form ax+b=c

The goal is to write the equation as The goal is to write the equation as variable=constant.variable=constant.

5x+6=26 ; solve for x by applying the 5x+6=26 ; solve for x by applying the Addition Property of equations to +6Addition Property of equations to +65x+6-6=26-65x+6-6=26-65x=20; divide both sides by 5; x=45x=20; divide both sides by 5; x=4 Check by replacing x with 4Check by replacing x with 4

3.3 GENERAL EQUATION PART 2 To solve an equation of the form To solve an equation of the form

ax+b=cx+dax+b=cx+d Apply the Addition Property of Equations Apply the Addition Property of Equations

twice and then the Multiplication Property twice and then the Multiplication Property of Equations to solve the unknown.of Equations to solve the unknown. 7a-5=2a-20; subtract 2a from both sides ; 7a-5=2a-20; subtract 2a from both sides ;

5a-5=-20; add 5 to both sides; 5a=-15; 5a-5=-20; add 5 to both sides; 5a=-15; divide both sides by 5; a=-3 and check.divide both sides by 5; a=-3 and check.

3.4 TRANSLATE AND SOLVE

Use the translation rules from chapter 2 and Use the translation rules from chapter 2 and solve the equations of one variable.solve the equations of one variable.

Consecutive Integer FormulasConsecutive Integer Formulas Consecutive Integers:n,n+1,n+2Consecutive Integers:n,n+1,n+2 Consecutive Even Integers; n,n+2,n+4Consecutive Even Integers; n,n+2,n+4 Consecutive Odd Integers; n,n+2,n+4Consecutive Odd Integers; n,n+2,n+4

SUM OF TWO NUMBERS WORD PROBLEMS Define the unknowns first.Define the unknowns first. Smaller number is x; Larger number is the Smaller number is x; Larger number is the

sum minus x (the smaller number).sum minus x (the smaller number). Translate and solve for the smaller number Translate and solve for the smaller number

first and then the larger number. Each first and then the larger number. Each problem must have two answers and add to problem must have two answers and add to equal the original sum.equal the original sum.

ADDITION AND SUBTRACTION OF POLYNOMIALS Monomial- a polynomial of one term.Monomial- a polynomial of one term. Binomial- a polynomial of two terms.Binomial- a polynomial of two terms. Trinomial- a polynomial of three terms.Trinomial- a polynomial of three terms. Quadnomial- a polynomial of four termsQuadnomial- a polynomial of four terms Descending Order- the exponents of the Descending Order- the exponents of the

variable decrease from left to right in the variable decrease from left to right in the answer.answer.

4.1 ADDING AND SUBTRACTING POLYNOMIALS

Addition of polynomialsAddition of polynomials Combine the like terms inside both sets of Combine the like terms inside both sets of

parentheses(same sign-ADD; different signs-parentheses(same sign-ADD; different signs-SUBTRACT).SUBTRACT).

Subtraction of polynomialsSubtraction of polynomials Multiply each term in the second polynomial by Multiply each term in the second polynomial by

–1(there is a minus sign in front of the –1(there is a minus sign in front of the parenthese) then combine the like terms in both parenthese) then combine the like terms in both polynomials.polynomials.

4.2 MULTIPLYING MONOMIALS Multiply the coefficients and add the like Multiply the coefficients and add the like

variable exponents.variable exponents. Simplifying powers of monomialsSimplifying powers of monomials

Distribute the outside exponent to each Distribute the outside exponent to each exponent in the monomial. Simplify the exponent in the monomial. Simplify the coefficient completely in the answer. coefficient completely in the answer. This is the only time exponents are This is the only time exponents are actually multiplied.actually multiplied.

4.3 MULTIPLICATION OF POLYNOMIALS Monomial times a polynomial.Monomial times a polynomial.

Multiply the monomial by applying the Multiply the monomial by applying the distributive property to each term inside the distributive property to each term inside the parentheses( the polynomial)parentheses( the polynomial)

Multiplying two polynomials.Multiplying two polynomials. Apply the distributive property by multiplying Apply the distributive property by multiplying

each term in the first polynomial by each term each term in the first polynomial by each term in the second polynomial and then combine the in the second polynomial and then combine the like terms. Place the answer un descending like terms. Place the answer un descending order.order.

TO MULTIPLY TWO BINOMIALS Use the FOIL method to multiply two binomials.Use the FOIL method to multiply two binomials. This is the simple application of the distributive This is the simple application of the distributive

property in an ordered method.property in an ordered method. F0IL METHOD;F- first terms are to be F0IL METHOD;F- first terms are to be

multiplied;O- outside terms are multiplied; I-multiplied;O- outside terms are multiplied; I-inside terms are multiplied;L-last terms are inside terms are multiplied;L-last terms are multiplied. Combine the middle two terms if multiplied. Combine the middle two terms if possible.possible.

MULTIPLYING BINOMIALS WITH SPECIAL PRODUCTS The Sum and Difference of two terms.The Sum and Difference of two terms.

Do FOIL; the middle two terms will Do FOIL; the middle two terms will cancel; the answer will be a binomial cancel; the answer will be a binomial with a minus sign between the terms.with a minus sign between the terms.

The Square of a binomial.The Square of a binomial. Do FOIL; the middle two terms will be Do FOIL; the middle two terms will be

the same so add them; the answer will be the same so add them; the answer will be a trinomial. a trinomial.

4.4 NEGATIVE EXPONENTS

Division of monomials.Division of monomials. To divide two monomials with the same To divide two monomials with the same

base, subtract the smaller exponent from base, subtract the smaller exponent from the larger exponent.the larger exponent.

Zero as an exponent.Zero as an exponent.If zero is the dominant exponent the If zero is the dominant exponent the

answer is always 1. answer is always 1.

RULES FOR SIMPLIFYING NEGATIVE EXPONENTS The negative exponent must be made positive by The negative exponent must be made positive by

moving it to the opposite place in the fraction. moving it to the opposite place in the fraction. This may be done first in the problem, but This may be done first in the problem, but especially in the answer. especially in the answer. If there is a like base in the numerator and If there is a like base in the numerator and

denominator and both exponents are negative denominator and both exponents are negative they must be switched and made positive; then they must be switched and made positive; then use division rules to simplify. use division rules to simplify.

MIORE RULES FOR NEGATIVE EXPONENTS If the bases are the same and one of the If the bases are the same and one of the

exponents is negative and one is positive, exponents is negative and one is positive, move the negative exponent to the positive move the negative exponent to the positive exponent and ADD the exponents together.exponent and ADD the exponents together.

When multiplying negative exponents, When multiplying negative exponents, combine the like base`s exponents together combine the like base`s exponents together using sign rules for addition and using sign rules for addition and subtraction. Make neg. exponents positive. subtraction. Make neg. exponents positive.

SCIENTIFIC NOTATION

In scientific notation, a number is expressed In scientific notation, a number is expressed as the product of two factors, the first as the product of two factors, the first number must be a number between one and number must be a number between one and ten(use of a decimal point may be needed), ten(use of a decimal point may be needed), and the other number a power of ten. and the other number a power of ten. To find the exponent in a number greater To find the exponent in a number greater

than one, count the place values after the than one, count the place values after the first number. first number.

MORE ON SCIENTIFIC NOTATION To write a decimal in scientific notation.To write a decimal in scientific notation.

Place a decimal point after the first Place a decimal point after the first number in the decimal.number in the decimal.

To write the power of ten, count the place To write the power of ten, count the place values from the decimal point to the first values from the decimal point to the first number in the decimal, this is the number in the decimal, this is the exponent.exponent.

4.5 DIVISION OF POLYNOMIALS TO divide a polynomial by a monomial.TO divide a polynomial by a monomial.

Divide each term of the Divide each term of the polynomial(numerator) by the polynomial(numerator) by the monomial.Simplify the expression.monomial.Simplify the expression.

TO DIVIDE POLYNOMIALS

The process for dividing polynomials is The process for dividing polynomials is similar to the one for dividing whole similar to the one for dividing whole numbers. The use of long division is the numbers. The use of long division is the method.method. Steps: Divide the like variable terms and Steps: Divide the like variable terms and

place the answer in the quotient. Multiply place the answer in the quotient. Multiply the quotient by each term on the outside the quotient by each term on the outside of the problem.of the problem.

STEPS FOR DIVISION

Step 3 is to subtract the products( change Step 3 is to subtract the products( change the sign of the second term and combine the the sign of the second term and combine the like terms).like terms).

The process starts over; divide, multiply, The process starts over; divide, multiply, and subtract.and subtract. If there is a remainder, write it as a If there is a remainder, write it as a

fraction.fraction.

5.1 GREATEST COMMON FACTOR Find the GCF of the coefficients which is Find the GCF of the coefficients which is

the largest number the numbers are the largest number the numbers are divisible by evenly.divisible by evenly.

Find the GCF of the variable parts by Find the GCF of the variable parts by choosing the variable part with the smallest choosing the variable part with the smallest exponent, but the variables must be in exponent, but the variables must be in common. common.

FACTORING BY GCF

Find the GCF of each term in the Find the GCF of each term in the polynomial and write it outside a polynomial and write it outside a parentheses.parentheses.

Divide each term in the polynomial by the Divide each term in the polynomial by the GCF and write it inside the parentheses. GCF and write it inside the parentheses. This is factoring by GCF.This is factoring by GCF.

FACTOR BY GROUPING

The polynomial must be a quadnomial(four The polynomial must be a quadnomial(four terms).terms).

Steps for factoring by grouping:Steps for factoring by grouping: Group the first two terms and the second Group the first two terms and the second

two terms with parentheses. The sign in two terms with parentheses. The sign in front of the third term is not inside the front of the third term is not inside the parentheses.parentheses.

STEPS FOR GROUPING

Find the GCF of each set of terms and Find the GCF of each set of terms and factor it out.factor it out.

To write the answer; write the common To write the answer; write the common binomial factor once and combine the binomial factor once and combine the GCF`s into one binomial and check the GCF`s into one binomial and check the sign of this binomial to make sure it is sign of this binomial to make sure it is right.right.

5.2 FACTOR BY EASY METHOD Factor out a GCF first, if possible.Factor out a GCF first, if possible. Find the signs of the binomials and Find the signs of the binomials and

place the correct variable in each place the correct variable in each binomial. binomial.

EASY METHOD

Find the factors of the last term Find the factors of the last term whose sum or difference equals the whose sum or difference equals the middle term. Write the correct middle term. Write the correct factors in the binomials.factors in the binomials.

5.3 TRIAL FACTORING

Factor out a GCF first, if possible.Factor out a GCF first, if possible. If the first term is 2 or greater, If the first term is 2 or greater,

factor by trial factors.factor by trial factors. Find the signs of the binomials Find the signs of the binomials

using the same rules as easy using the same rules as easy method.method.

TRIAL FACTORING

Find the factors of the first and last Find the factors of the first and last terms and place them in a chart.terms and place them in a chart.

Do outer and inner FOIL with the Do outer and inner FOIL with the factors. The answer must match the factors. The answer must match the middle term. Write the factors in middle term. Write the factors in the correct binomials and check. the correct binomials and check.

5.4 SPECIAL FACTORING

To factor the difference of two To factor the difference of two squares.squares.The problem must be a binomial The problem must be a binomial

with a negative sign.with a negative sign.The signs of the binomials will The signs of the binomials will

be (+) and (-).be (+) and (-).

THE DIFFERENCE OF TWO SQUARES Find the perfect squares of both Find the perfect squares of both

terms and set the up as the terms and set the up as the difference of two squares.difference of two squares.

Write the terms twice, once in each Write the terms twice, once in each binomial.binomial.

PERFECT- SQUARE TRINOMIALS This method may be used as a This method may be used as a

shortcut to trial factoring.shortcut to trial factoring. Criteria:Criteria:

Must be a trinomial with a (+) Must be a trinomial with a (+) sign in front of the last term.sign in front of the last term.

CRITERIA FOR SPECIAL FACTORING

The first and last terms must The first and last terms must have perfect squares.have perfect squares.

Multiply the perfect squares Multiply the perfect squares together twice and add them. The together twice and add them. The answer must match the middle answer must match the middle term or factor by another term or factor by another method. method.

5.5 SOLVING EQUATIONS BY FACTORING The Principle of Zero Products The Principle of Zero Products

states that if the product of two states that if the product of two factors is zero, then at least one of factors is zero, then at least one of the factors must be zero.the factors must be zero.

If a x b = 0, then a =0 or b =0. If a x b = 0, then a =0 or b =0.

QUADRATIC EQUATION

A Quadratic Equation is in A Quadratic Equation is in standard form when the standard form when the polynomial is in descending order polynomial is in descending order AND equal to zero.AND equal to zero.

Factor and solve. Each problem Factor and solve. Each problem will have two answers.will have two answers.

6.1 TO SIMPLIFY A RATIONAL EXPRESSION Factor the numerator and Factor the numerator and

denominator.denominator. Divide by the common factors.Divide by the common factors. Be careful with the sign of the Be careful with the sign of the

simplified answer.simplified answer.

TO MULTIPLY RATIONAL EXPRESSIONS Factor ALL numerators and Factor ALL numerators and

denominators.denominators. Divide by the common factors.Divide by the common factors. Multiply the numerators.Multiply the numerators. Multiply the denominators.Multiply the denominators.

TO DIVIDE RATIONAL EXPRESSIONS

Change division to multiplication Change division to multiplication and invert the second fraction.and invert the second fraction.

Follow the steps for multiplication Follow the steps for multiplication to simplify the problem.to simplify the problem.

Be careful with the sign of the Be careful with the sign of the answer.(Multiplying and dividing)answer.(Multiplying and dividing)

6.2 FINDING THE LCM

Factor the denominators first.Factor the denominators first. To find the LCM:To find the LCM: What is the greatest number of What is the greatest number of

times a term(monomial) occurs or times a term(monomial) occurs or a set of terms(binomial) occurs?a set of terms(binomial) occurs?

ADDITION AND SUBTRACTION Factor the denominators.Factor the denominators. Find the LCM of the denominators Find the LCM of the denominators

and place them under one fraction and place them under one fraction bar.bar.

Place the fractions in higher terms.Place the fractions in higher terms.(Divide and Multiply)(Divide and Multiply)

ADDITION AND SUBTRACTION STEPS Combine like terms in the Combine like terms in the

numerator.numerator. Simplify the answer(factor and Simplify the answer(factor and

cancel).cancel).

6.4 COMPLEX FRACTIONS

Find the LCM of the denominators of Find the LCM of the denominators of the fractions in the numerator and the fractions in the numerator and denominator.denominator.

Multiply the LCM by every term in the Multiply the LCM by every term in the numerator and denominator.numerator and denominator.

Simplify the answer(factor and cancel).Simplify the answer(factor and cancel).

6.5SOLVING EQUATIONS WITH FRACTIONS Find the LCM of the denominators.Find the LCM of the denominators. Multiply the LCM by every term in Multiply the LCM by every term in

the problem.the problem. Solve and check( if any Solve and check( if any

denominators equal zero the denominators equal zero the answer is NO SOLUTION).answer is NO SOLUTION).

6.6 RATIO AND PROPORTION

Cross multiply and solve for the Cross multiply and solve for the unknown.unknown.

In the word problems, make sure In the word problems, make sure that the rates are set up with like that the rates are set up with like units on top and like units on the units on top and like units on the bottom.bottom.

6.7 LITERAL EQUATIONS

A literal equation is an equation A literal equation is an equation that has more than one variable.that has more than one variable.

Use the Addition and Use the Addition and Multiplication Properties to help Multiplication Properties to help solve for one of the variables.solve for one of the variables.

10.1 RADICAL EXPRESSIONS

A square root of a number x is a A square root of a number x is a number whose square is x.number whose square is x.

The square root of 16 is 4.The square root of 16 is 4. The number 4 is considered the The number 4 is considered the

perfect square of 16.perfect square of 16.

PRODUCT PROPERTY

If the number under the radical If the number under the radical does not have a perfect square, does not have a perfect square, apply the Product Property of apply the Product Property of Square Roots.Square Roots.

Find the first number that has a Find the first number that has a perfect square that goes into the perfect square that goes into the number evenly.number evenly.

625 = 25=

360 = 1036Find the square root of 36 which is 6 and leave 10 under the radical sign.

MORE ON THE PRODUCT PROPERTY Write the perfect square on the Write the perfect square on the

outside of the radical and the outside of the radical and the number that does not have a number that does not have a perfect square on the inside the perfect square on the inside the radical.radical.

TO SIMPLIFY VARIABLE RADICAL EXPRESSIONS If there is a variable under the If there is a variable under the

radical, find the perfect square by radical, find the perfect square by dividing the exponent by 2 and dividing the exponent by 2 and write the answer.write the answer.

10.2 ADDITION AND SUBTRACTION OF RADICALS

Apply the Product Property to Apply the Product Property to each term. If the terms under the each term. If the terms under the radical are the same, combine the radical are the same, combine the like terms outside the radical. like terms outside the radical.

10.3 MULTIPLICATION AND DIVISION OF RADICALS Multiply the terms under the Multiply the terms under the

radicals and place the answer under radicals and place the answer under one radical.one radical.

Apply the Product Property. Apply the Product Property.

MULTIPLYING RADICALS

If there are parentheses then apply If there are parentheses then apply the Distributive Property or FOIL the Distributive Property or FOIL and then combine like terms if and then combine like terms if possible. possible.

Conjugates are the sum and Conjugates are the sum and difference of two terms. difference of two terms.

DIVISION OF RADICALS

Rewrite the radical expression as Rewrite the radical expression as the quotient of the square roots.the quotient of the square roots.

Apply the Product to each term.Apply the Product to each term. Simplify and write the answer.Simplify and write the answer. The answer cannot have radical in The answer cannot have radical in

the denominator.the denominator.

DIVISION OF RADICALS

If the denominator has a radical, If the denominator has a radical, the process is called rationalizing the process is called rationalizing the denominator.the denominator.

Multiply the denominator by both Multiply the denominator by both the numerator and denominator.the numerator and denominator.

Simplify what is left.Simplify what is left.

7.1 GRAPHING

To graph the ordered pair (2,3), the To graph the ordered pair (2,3), the 2 is plotted along the x-axis and 3 2 is plotted along the x-axis and 3 is plotted on the y-axis.is plotted on the y-axis.

The origin is 0 on the graph.The origin is 0 on the graph. 2 is the x-coordinate and 3 is the y-2 is the x-coordinate and 3 is the y-

coordinate.coordinate.

RELATIONS AND FUNCTIONS A relation is any set of ordered A relation is any set of ordered

pairs.pairs. The domain is the set of first The domain is the set of first

coordinates.coordinates. The range is the set of second The range is the set of second

coordinates.coordinates.

FUNCTIONS

A function is a relation in which no A function is a relation in which no two ordered pairs that have the two ordered pairs that have the same first coordinate have different same first coordinate have different second coordinates.second coordinates.

GRAPHING EQUATIONS OF THE FORM Y=MX+B In this section we learn to graph In this section we learn to graph

equations three different ways.equations three different ways. Define a set of three values for x Define a set of three values for x

and solve the equation, graph the and solve the equation, graph the three sets of ordered pairs and three sets of ordered pairs and connect them with a straight line.connect them with a straight line.

9.2 ADDITION AND MULTIPLICATION PROPERTIES

To solve an inequality, apply the To solve an inequality, apply the same properties that were used to same properties that were used to solve equations of one variable.solve equations of one variable.