Fundamentals of Algebra Notes (To Accompany Hawkes Textbook) Kent M. Willis RaptorMath.com

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1 Fundamentals of Algebra Notes (To Accompany Hawkes Textbook) Kent M. Willis RaptorMath.com June 3, 2012

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3 Contents 1 Real Numbers Introduction to Real Numbers Addition with Real Numbers Subtraction with Real Numbers Multiplying and Dividing Real Numbers Exponents, Prime Numbers, and LCM Multiplication and Division with Fractions Fractions: Definitions Reducing Fractions to Lowest Terms Converting a mixed number to fraction form Converting an improper fraction to a mixed number Multiplying Fractions Dividing Fractions Addition and Subtraction with Fractions Expanding Fractions Adding Fractions Subtracting Fractions Word Problems with Fractions Order of Operations PEMDAS Average Averages of trickier numbers Properties of Real Numbers

4 4 CONTENTS 2 Algebraic Expressions and Linear Equations Simplifying and Evaluating Algebraic Expressions Definitions: Variables, Expressions, and Equations Evaluating Expressions Checking Solutions to Equations Equations vs Expressions Definitions: Terms, Coefficients, and Like Terms Combining Like Terms Simplifying Expressions Translating English Phrases and Algebraic Expressions Translating Words to Expressions using Variables Translating Algebraic Expressions into Words Solving Linear Equations: x + b = c and ax = c Addition Property of Equality Solving Equations with the Variable on Both Sides Simplifying an Equation Before Solving Solving Equations, Step by Step (First cut) Multiplication Property of Equality Division Does Not Always Come Out Even The Pesky x Simplifying Before Solving Solving Equations, Step by Step (Complete) Solving Linear Equations: ax + b = c Combining addition and multiplication: ax + b = c Unusual cases Solving Linear Equations: ax + b = cx + d Solving Linear Equations with the variable on both sides Clearing Fractions: Solving a b x + c d = m n Clearing Decimals: Solving 0.0a(x b) + 0.c = 2.d Equations with Infinitely Many Solutions (Identities) Equations with No Solution (Contradictions) Introduction to Problem Solving Procedure: Working Word Problems (A General Procedure)

5 CONTENTS Unknown Number Problems Consecutive Integer Problems Consecutive Even (or Odd) Integer Problems Applications with Percent Converting between percent, decimals, and fractions Percentage word problems Discounts, markups, and similar things Percent profit Formulas and Linear Inequalities Working with Formulas Definition: Formula Finding the Value of the Last Variable Solving a Formula for a Certain Variable Formulas in Geometry Perimeter Area Volume Applications: Distance-Rate-Time, Interest, Average Procedure: Using a table to solve word problems Applications with averages Weighted Averages Linear Inequalities Inequalities Intervals Open and Closed Intervals Intervals that go on forever Graphing Intervals Interval Notation Solving Inequalities Three-part Inequalities Absolute Value Equations and Inequalities Absolute Value Equations Solving absolute value inequalities

6 6 CONTENTS 4 Linear Equations and Functions in Two Variables 79 5 Exponents and Polynomials Exponents Exponent Review A Few New Reminders Product Rule for Exponents Share the Power Double Power Combinations Exponents and Scientific Notation Quotient Rule for Exponents The Zero Exponent Negative Exponents Changing Between Negative and Positive Exponents Combinations Scientific Notation Putting a number into Scientific Notation Putting a number into Normal Notation Multiplying and Dividing In Scientific Notation Introduction to Polynomials Definitions Degree of a Polynomial Evaluating Polynomials Why synthetic evaluation is possible Synthetic evaluation: a nifty alternative Addition and Subtraction with Polynomials Adding Polynomials Subtracting Polynomials Working with more than one variable Multiplication with Polynomials Multiplying One Term times One Term (One-by-One) Multiplying One Term times Two or More Terms (One-by-N) Multiplying Two or More Terms times Two or More Terms (M-by-N)

7 CONTENTS Multiplying More Things Together Special Products of Binomials Squaring a binomial (Double-Stuff Shortcut) Multiplying a Sum and Difference (No-Stuff Shortcut) Finding Greater Powers of a Polynomial Division with Polynomials Dividing one term by one term (one-by-one) Dividing two or more terms by one term (n-by-one) Dividing two or more terms by two or more terms (m-by-n) Synthetic division Factoring and Solving Quadratic Equations Greatest Common Factor (GCF) and Factoring by Grouping (White and Yellow Belts) Prime Factorization Finding the GCF of Natural Numbers Finding the GCF of Terms with Variables White Belt: Factoring out the Greatest Common Factor (GCF) Yellow Belt: Factoring by Grouping Unfactorable polynomials Some tips for oddball problems: Factoring Trinomials: x 2 + bx + c (Orange Belt) The Key Number Method Special notes regarding negatives Factoring trinomials with two variables Factoring Trinomials: ax 2 + bx + c (Purple Belt) Texas Box Method A-C method Special Factorizations Green Belt: Factoring the Difference of Squares Blue Belt: Factoring the Sum or Difference of Cubes Additional Factoring Practice Red Belt: Special Cases and Mixed Factoring Factoring Perfect Square Trinomials Mixed Factoring

8 8 CONTENTS 7 Rational Expressions Multiplication and Division with Rational Expressions The Basics of Rational Expressions Evaluating Rational Expressions The Zero Factor Law (ZFL) Finding Bad Values for Rational Expressions Canceling Common Factors Using Factoring to Reduce Rational Expressions Reducing with Opposites Equivalent Forms Multiplying Rational Expressions Working with Parentheses Factoring First Before Reducing Dividing Rational Expressions Addition and Subtraction with Rational Expressions Least Common Denominator (LCD) Expanding Rational Exressions to a Given Denominator Adding Rational Expressions with matching denominators Adding Rational Expressions with different denominators Subtracting Rational Exressions Complex Fractions Method I (Fraction over Fraction) Method II (Addition or Subtraction within one floor) A Geometric formulas 163 A.1 Two-Dimensional Shapes A.2 Three-Dimensional Shapes

9 Chapter 1 Real Numbers This chapter is almost entirely a review of arithmetic, especially some of the trickier parts of it. You should make sure that you not only know how to work with fractions and negative numbers, but also that you are confident that you are able to do it right without using notes or a calculator. Remember, if you are struggling with arithmetic, work hard on it now, and get lots of practice. Then when the material gets harder as we progress through the course, you will not have to struggle with this part any more. You will then be able to concentrate better on the new material. Some students underestimate the importance of being able to do these operations well, including actually getting the right answer. Then, later on, they find themselves on the wrong side of an avalanche when they are expected to actually use the techniques. Another word of caution: don t rely too much on your calculator. You should practice doing operations by hand. Why? Because calculators will do what you tell them to do, but not what you merely intend for them to do. You can fat-finger and press the wrong button, or you can forget a parenthesis. The calculator will then very quickly tell you the wrong answer. The only way that you will have a clue that anything is wrong before you hand in your work is if you kinda know what to expect, and are familiar with how things should be; and the only way to get that is to do the work by hand. Then you can get a sense about what is crazy and needs to be checked over. Don t trust your calculator to be infallible. Your calculations are only as good as you make them, and it s way too easy to make a mistake on a calculator. Also, when we get into the real algebra, you will need to know how to do things by hand, since a calculator can t handle variables. Ask a calculator to add x and see what happens. It can t do it. But it s easy and quick to do when you really understand how to add fractions. 1.1 Introduction to Real Numbers This section will be skipped for this course, but practice problems are available in the software: 1.1a The Real Number Line and Inequalities 1.1b Introduction to Absolute Values 9

10 10 CHAPTER 1. REAL NUMBERS 1. Graph the odd integers between 5 and 10. Skills Check 2. Graph {x N x < 4}. 3. Which of these numbers are rational numbers: { 7, } 2, 0, 4.2, π, 17 4, Put in a <, >, or = symbol to make this statement true: Put in a <, >, or = symbol to make this statement true: 6 6 3

11 1.2. ADDITION WITH REAL NUMBERS Addition with Real Numbers This section will be skipped for this course, but practice problems are available in the software: 1.2 Addition with Real Numbers 1.3 Subtraction with Real Numbers This section will be skipped for this course, but problems are available in the software: 1.3 Subtraction with Real Numbers Skills Check 1. Simplify by adding or subtracting: 4 + ( 6) 2. Simplify by adding or subtracting: 4 + ( 6) 3. Simplify by adding or subtracting: 4 ( 6) 4. Simplify by adding or subtracting: 4 ( 6)

12 12 CHAPTER 1. REAL NUMBERS 1.4 Multiplying and Dividing Real Numbers This section will be skipped for this course, but practice problems are available in the software: 1.4 Multiplication and Division with Real Numbers 1. Simplify by multiplying: 4( 6) Skills Check 2. Simplify by multiplying: 4( 6) 3. Simplify by multiplying: 4(6) 4. Simplify by multiplying: 4(6) 5. Simplify by multiplying and dividing: 3 (2) 6( 5)

13 1.5. EXPONENTS, PRIME NUMBERS, AND LCM Exponents, Prime Numbers, and LCM This section will be skipped for this course, but practice problems are available in the software: 1.5 Exponents, Prime Numbers, and LCM 1. Rewrite using exponents: Skills Check 2. Find the prime factorization of Find the LCM of: 20x, 30xy, and 40x 2

14 14 CHAPTER 1. REAL NUMBERS 1.6 Multiplication and Division with Fractions Practice problems are available in the software: 1.6a Reducing Fractions 1.6b Multiplication and Division with Fractions Fractions: Definitions Proper fractions have a smaller number upstairs (in the numerator) than they have downstairs (in the denominator). Improper fractions are top-heavy, with a larger number upstairs than downstairs. Improper makes it sound like they are bad. But they are good and useful. In Algebra, we prefer improper fractions over mixed numbers. Examples 1 2 and 3 5 are proper fractions; 5 3 and 10 7 are improper fractions. Mixed numbers are an integer and a fraction added together, bonded together as a single item. Mixed numbers are useful in recipes and construction projects, but not very useful in Algebra. Example = ( ) Product the result of multiplication; the answer Example: 12 is the product of 2 and 6 Factor numbers that can be multiplied by another number to get a certain product Example: 1, 2, 3, 4, 6, and 12 are all factors of 12 Divisible One number is divisible by a second number if the first number divided by the second number has a remainder of 0. A number is always divisible by its factors. Examples 12 is divisible by 4, because 12 4 = 3 R 0 (no remainder) 12 is not divisible by 5, because 12 5 = 2 R 2.

15 1.6. MULTIPLICATION AND DIVISION WITH FRACTIONS Reducing Fractions to Lowest Terms Lowest Terms means that all common factors in the upstairs and downstairs of a fraction have been canceled. Canceled factors become 1, not 0 or just nothing. There are several valid methods to use to reduce a fraction to lowest terms. If you have a favorite, use it. If you don t, here is one that is very thorough and always works. There are quicker ways that are equally valid. Reducing fractions The most thorough method 1. Find the prime factorization of the upstairs and of the downstairs. 2. Cross out any prime factor that appears on both floors. 3. If all the factors upstairs are crossed out, leave a 1 upstairs. 4. If all the factors downstairs are crossed out, just write the upstairs as a plain number (no longer a fraction). Example Reduce to lowest terms = = 3 2 Example Reduce to lowest terms = = 5 3 Example Reduce to lowest terms = = Converting a mixed number to fraction form This is standard procedure whenever we see a mixed number. It is difficult to do math with them, so the first thing we do with a mixed number is convert it to a fraction. The upstairs of the resulting fraction is the tricky part. The downstairs doesn t change. 1. Multiply the downstairs of the fraction part times the whole number, then add the upstairs to that. This is the upstairs of the resulting fraction. 2. The downstairs of the resulting fraction is exactly the same as the downstairs of the fraction you started with. Example: Convert to fraction form 1. upstairs = = downstairs stays 4. Answer: 11 4

16 16 CHAPTER 1. REAL NUMBERS Watch out for negative mixed numbers. When converting to fraction form, leave the minus sign for last. First convert it like a positive mixed number, then put the negative sign in front of the fraction. Example: Convert to fraction form 1. ignore the negative sign for now, so the upstairs = = downstairs stays 7; apply the negative sign now. Answer: Converting an improper fraction to a mixed number This is not an important skill in an Algebra class. You will never need to do this for Algebra at all. But in case you are working on a project where it would be nice, here is how to do it. Optional Procedure. Conversion using paper and pencil. This procedure is not necessary for you to learn to pass the course. 1. Divide the upstairs by the downstairs. 2. The quotient part (on top of the bracket) is the whole number part of the answer. 3. The remainder is the new upstairs part of the fraction. 4. The downstairs does not change. Conversion using a calculator. 1. Divide the upstairs by the downstairs. 2. The part before the decimal is the whole number part of the answer. The digits after the decimal can be ignored. 3. Take the whole number part times the downstairs and subtract that from the upstairs. This is the remainder, which becomes the new upstairs part of the fraction. 4. The downstairs does not change. Example Convert 13 5 to a mixed number = 2 R 3, so 2 is the whole number part, and 3 is the new upstairs. Answer: Multiplying Fractions Simple method. This is the easiest method to understand, but it can cause you to work with large numbers and do more work in the long run.

17 1.7. ADDITION AND SUBTRACTION WITH FRACTIONS Multiply tops to get the upstairs part of the answer. 2. Multiply bottoms to get the downstairs. 3. Reduce if possible. Example = Cancel First method. You can try to cancel factors before multiplying to avoid getting large numbers to reduce later. This method usually saves work in the long run. 1. Find the prime factorization of each fraction s upstairs and downstairs. 2. Cross out common factors from any upstairs and any downstairs (same or different fraction, remembering they cancel to 1). 3. Multiply all the factors upstairs to get the upstairs of the answer. 4. Multiply all the factors downstairs to get the downstairs of the answer. 5. No further reducing will be required (if you found all the common factors). Example = = Dividing Fractions Nobody ever directly divides fractions without using a calculator. It would be really messy. Instead, we multiply by the reciprocal. Invert the divisor and then multiply. The divisor is the number immediately after the symbol. Flip it upside down (upstairs goes down, and downstairs goes up), change the sign to multiplication, and then multiply! Example = = Addition and Subtraction with Fractions Practice problems are available in the software: 1.7 Addition and Subtraction with Fractions

18 18 CHAPTER 1. REAL NUMBERS Expanding Fractions This is opposite of reducing fractions. If we multiply the top and bottom of a fraction by the same number and do not reduce, the numbers get bigger, but the value of the fraction doesn t change. Why? Because we are really just multiplying the fraction by 1! Example Expanding = = 3 6 several times: = = 10 5 Procedure Expanding a fraction to a certain denominator 1. Find what you need to multiply the downstairs by to meet the goal (goal starting value). 2. Multiply upstairs and downstairs by that number. Example Expand 2 3 to a fraction with denominator = 4, so we have to multiply top and bottom by = Adding Fractions Don t get this confused with the method for multiplying fractions! These methods are different and very specific! Procedure Adding fractions with matching bottoms If the downstairs parts of two added fractions match, simply: 1. Add the upstairs parts together 2. Carry the downstairs over without change 3. Then reduce if possible Example = = 5 7 Notice the downstairs part remains the same, unless the answer can be reduced. Example = 4 8, which reduces to 1 2 Procedure Adding Fractions with Different Denominators Don t take a shortcut to perdition = 2 5!!!

19 1.7. ADDITION AND SUBTRACTION WITH FRACTIONS Find a common denominator (a) Find the LCM (?? on page??) of the downstairs number. This is the smallest common denominator possible. (b) If you get stuck, just multiply all the downstairs parts together and you will have a common denominator (maybe not the smallest one). You will probably have more reducing to do, but it ll get the job done. 2. Expand each fraction to have that common denominator. 3. Now that the fractions have the same downstairs, just add the tops and then reduce if possible. Example Add A common denominator would be 6 2. Expanding each fraction to have downstairs = = = Adding tops, we get: = 5 6 Example Add A common denominator would be Expanding each fraction to have downstairs = = = Adding tops, we get: = Subtracting Fractions Subtracting fractions uses the same rules as adding fractions. Except you subtract the tops Word Problems with Fractions Dealing with fractions inside word problems is a double-whammy. Instead of freaking out, it may be helpful to first figure out, "How would I do this problem with regular numbers?" Then substitute the fractions back into place. That way you can deal with one freaking at a time. Procedure Solving word problems 1. What is the question? Make it short and simple. 2. What operations (addition, subtraction, etc.) do I need to do to answer such a question?

20 20 CHAPTER 1. REAL NUMBERS 3. Gather the relevant numbers from the story. 4. Do the operations. 5. Check your answer for real life. Example A gallon of paint covers 500 ft 2. To paint his house, Hercomer needs enough paint to cover 4200 ft 2. How many gallons of paint should he buy? 1. The question, short and sweet, is "how many gallons?" 2. The operation needed is division, because we have a big job divided up among several gallons. We need to take the total required and divide by how much each gallon can do = 8.4 or Hercomer can t buy part of a can of paint. (Plus he might spill some.) So real life would demand that he buy 9 gallons. Answer: 9 gallons. 1. Reduce 14 to lowest terms. 21 Skills Check 2. Perform the indicated operations: (a) (b) (c) (d) A certain rectangle is 17/8 inches by 23/4 inches. (a) What is the perimeter of the rectangle? (b) What is its area?

21 1.8. ORDER OF OPERATIONS Order of Operations Practice problems are available in the software: 1.8 Order of Operations PEMDAS While you are practicing on these problems, until you have the procedure down pat, it s a good idea to have a little thing that s easy to remember in your head. Some people just use the acronym PEMDAS and have no trouble with it. Others like Please Excuse My Dear Aunt Sally as a mnemonic device. My favorite one for a couple of reasons, is the classic Penguins Eat Mustard & Dumplings, Apples & Spice. My family has a special fondness for penguins. However, it also uses the ampersand (& symbol) very appropriately, since it ties M and D together to the same level, and also ties the A and S together at the same level. Operation is one of the basic four things that we can do with numbers: addition, subtraction, multiplication, and division. By extension, exponents fit in as an operation also, since exponents are a shortcut for multiplication. Parentheses are not an actual operation, but they affect the order that we do operations in. Example With , if we do the multiplication first, we get 10 as the answer. But if we do the addition first, we get 14 as the answer. Which is right? Procedure PEMDAS: Please Excuse My Dear Aunt Sally, or Penguins Eat Mustard & Dumplings, Apples & Spice 1. P = parentheses of any shape ( ) [ ] { } and also implied parentheses. (Things like fraction bars, absolute values, and radicals have their own job to do, but they also imply parentheses. More to come on that later.) 2. E = exponents (these affect only what they touch) 3. M,D = Multiplication and Division (these are at the same level) 4. A,S = Addition and Subtraction (these are at the same level) 5. Work line by line so that each line is complete. 6. For operations at the same level, work the one on the left first. 7. Work from the innermost parentheses outward.

22 22 CHAPTER 1. REAL NUMBERS 8. Remove parentheses only when the last operation inside is complete. 9. Put in a multiplication symbol if there is no operator between a number and the parentheses. Example Getting back to , the multiplication must be done first, since M is earlier in the list than A. So we can write: (multiply 2 3) (add) 10 Important! Modern calculators have the order of operations built in, but it can sometimes be tough to enter everything exactly right (especially with parentheses, and even more especially with implied parentheses). It s best to do the job by hand and then check it with a calculator. If they agree, you re probably correct. Example Simplify 9[(4 + 8) 3] 9[(4 + 8) 3] (add in innermost parentheses) 9[12 3] (subtract 12-3 in parentheses, insert multiplication symbol) Example (divide 16 4) (multiply 4 3) (subtract 5-2) Example 2(7 + 8) Note. The fraction bar is one case of implied parentheses. The entire upstairs is enclosed in implied parentheses, and the whole downstairs likewise. So the upstairs downstairs will be the last operation. 2(7 + 8) + 2 (add on top in parentheses) (multiply 2 15 on top, and 3 5 on bottom) (add on top, and on bottom) (do the division last) 16 2

23 1.8. ORDER OF OPERATIONS Average There are actually several different things that can be called the average of a group of numbers. Most commonly, the word average refers to a thing technically known as the arithmetic mean. Other things that can be called an average include: geometric mean, median, and mode. These other types will not be covered here. The average of two numbers is really the number that is half-way between them. So it is easy to find the average of 2 and 4; since 3 is the middle value between them, the average of 2 and 4 is 3. But what about when there are more than a pair of numbers, so you can t tell what s half-way between whom? Here is a procedure that works for any size list: Procedure Finding the average of a list of numbers 1. Add all the numbers together (to get the sum ). 2. Count how many numbers there are in the original list (to get the count ). 3. Divide the sum from step 1 by the count from step 2. Example Find the average of 6, 10, and = 27, so the sum is There are 3 numbers in the list, so the count is = 9 Answer: Averages of trickier numbers If you need to find the average of fractions or mixed numbers, the procedure doesn t change at all. The only difference is that you will need to add and divide with fractions. You still add all the numbers and then divide by the count. If you need to find the average of numbers, some of which are negative, the procedure doesn t change either. The big deal here is that the average can be negative or zero. Sometimes people freak out when the answer comes out zero, as if there is some law that says you can t get 0 for an answer. Zero is a perfectly acceptable answer, as long as you did your math correctly. Example Find the average of 3 and Find the sum: = = Determine the count: There are 2 numbers in the list. 3. Divide the sum by the count: = = 1 4 Answer: 1 4

24 24 CHAPTER 1. REAL NUMBERS Skills Check 1. Use the order of operations to simplify: 9 3 [2 (3 2)] 2. Use the order of operations to simplify: Use the order of operations to simplify: ( ) Find the average of 14, 23, 0, and Mergatroid took English, Speech, Algebra, and Chemistry his first semester of college. Use a weighted average to calculate his GPA (with A = 4, B = 3, C = 2, and D = 1). Course Credits Grade English 3 B Speech 2 C Algebra 3 B Chemistry 4 A

25 1.9. PROPERTIES OF REAL NUMBERS Properties of Real Numbers This section will be skipped for this course, but practice problems are available in the software: 1.9 Properties of Real Numbers Skills Check 1. Complete the right side of the equation to illustrate the given property: (a) Commutative property of addition: = (b) Associative property of addition: (5 + x) + 6 = (c) Commutative property of multiplication: 5 2 = (d) Associative property of multiplication: (15 9) 13 = (e) Distributive property: 5 (2 + 7) = (f) Additive inverse: 4 + ( 4) = (g) Multiplicative inverse: = 2. Name the property shown in: (3 + a) + 4 = 4 + (3 + a)

26 26 CHAPTER 1. REAL NUMBERS

27 Chapter 2 Algebraic Expressions and Linear Equations This chapter is where we begin really working with Algebra itself. The letter x will begin to show up frequently, standing for an unknown number. Sometimes we want to find out what x is equal to, and sometimes we have no way of knowing what it is, but we just work with it and simplify things; in those cases, the answer still has the x in it. Often students are bothered by this, and try very hard to do something about it. Unfortunately, these attempts always end up with a wrong answer. You can t assume that x has some value and stick it in for x. It just doesn t work. The best advice for students struggling with this chapter is to practice as close to daily as possible. It is much better to work on your homework for at least 15 minutes every day than to have one marathon session every week or two. Also do your best to be ready for an exam every time you come to class. Keeping all the little bits and pieces straight, knowing when to do what, and remembering the rules for each kind of game is so very much easier when you refresh your skills every day. 2.1 Simplifying and Evaluating Algebraic Expressions Practice problems are available in the software: 2.1a Variables and Algebraic Expressions 2.1b Simplifying Expressions 2.1c Evaluating Algebraic Expressions Definitions: Variables, Expressions, and Equations Variable a letter (like x) that stands for an unknown number. 27

28 28 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS Watch out! DO NOT assume x = 1 when you don t know what else it could be. Many people get confused on this point. Assume it is some number we don t know. Then you ll be OK. Constant a number whose value never changes. Like 2. Nice and solid. Sometimes we use a letter for a constant, too, but usually these are Greek letters like π. Its value never changes, either. It s just hard to write an infinite number of digits. Expression a combination of one or more variables and/or constants with optional operators thrown in. Pretty much anything goes for these, except = and the inequality symbols. Equation a statement that two expressions are equal. So an equation MUST have an = symbol. Inequality a statement that one expression is or may be unequal to another expression. So an inequality MUST have one of these symbols: <, >,,, or = Evaluating Expressions One type of problem you will be given is to evaluate an expression. That means to find the value of whatever they give you. Obviously, there must be one or more variables in the expression or the game would be pointless. But you can never ever know what an expression is equal to unless you are told the value of every variable in the expression. So, in a problem of this sort, you are told a value to use for each variable. Just plug it in where the variable is (in place of the variable), and then simplify, using the order of operations. Example Evaluate 16p if p = 3. Important! Remember that 16p means 16 p. Since p = 3, putting 3 in place of p makes 16 3 = 48. Example Evaluate 2p 3 if p = (remember to use the order of operations correctly) Example Evaluate 2x 2 + y 2 if x = 6 and y = 9. 2(6) 2 + (9)

29 2.1. SIMPLIFYING AND EVALUATING ALGEBRAIC EXPRESSIONS Checking Solutions to Equations Solution A solution to an equation is a value that you can put in place of the variable (just like evaluating above), simplify, and get a true statement like 2 = 2. Procedure Checking a possible solution 1. Use a given number to evaluate the expression on each side of the equation. Remember to use the order of operations correctly. 2. If you get a statement that is always true, like 2 = 2, the given number is a solution. 3. If you get a false statement, like 4 = 3, the given number is NOT a solution. Note. For now, we will be working with equations that have exactly one solution. Later, we will find cases where there is no solution, and other times that there are multiple solutions. You can always use this method to check any potential solution, but if you find one solution, it does not mean that there are no more solutions. There could always be another one lurking around somewhere... Example Is 2 a solution of 8p 11 = 5? = 5? = 5? 5 = 5? True! Yes, 2 is a solution. Example Is 4 a solution of 9m 6 = 32? = 32? 36 6 = 32? 30 = 32? False! No, 4 is not a solution Equations vs Expressions Important! A few things to remember: Equations have an equals symbol =, and you can generally solve them. Expressions do not have an equals symbol; you can sometimes simplify them, but you can never solve them. You can only evaluate expressions if you are given a value for each variable in it. If you simplify an expression, you will generally have the variable stay there in the mix. It is the oddball problem where you get a plain number for the answer.

30 30 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS Example Determine if this is an equation or an expression: 3x 1 5 Answer: expression, since there is no equals sign Example Determine if this is an equation or an expression: 3x 1 = 5 Answer: equation, since there is an equals sign Definitions: Terms, Coefficients, and Like Terms Term A term is collection of one or more constants or variables that are multiplied together. Addition and subtraction separate terms from each other. Signs are kept with the term that follows the sign. Example The following list are all single terms: 2, x, 2x, 2x 3, 7x 2 y 3, and 2longandtricky Example How many terms are in 2x 3y + 7 4? There are four terms: 2x, 3y, 7, and 4. When you keep the sign with the term that follows it, like 4 in the previous example, think of all the terms as being added together. Coefficient The number part of a term. Leave off all the variables and their exponents. If there is no number part, the coefficient is 1, because x = 1 x = 1x. This does NOT mean that x can be assumed to be equal to 1. That would be false, false, false, and very false. Example What is the coefficient of 3x 3 y? Answer: 3 Like Terms Like terms are terms that match exactly on the variable part. Same letters, same exponents. Different order is OK. If two terms do not match that exactly, they are called unlike terms. Example Are the following pairs of terms like or unlike? 2a and 3a 2a and 3ab 2a and 3a 2 2ab and 3ba 2a and 3 2 and 3 a and 3a like terms unlike, different letters unlike, different exponents like terms, order does not matter unlike, different letters like terms like terms Combining Like Terms We can combine like terms by adding their coefficients, and leaving the variable part as it is.

31 2.1. SIMPLIFYING AND EVALUATING ALGEBRAIC EXPRESSIONS 31 Example Combine like terms: 2x + 7x 2x + 7x = 9x Notice it is not 9x 2 or other deviation. It is like saying, 2 apples plus 7 apples is 9 apples. Not 9 apples squared. Remember the Double Roll Rule: 1x + 1x = 2x. What if there is no coefficient? Remember x = 1 x = 1x, so the coefficient is defined to be 1 every time there is no coefficient visible. Example Combine like terms: 3x + x 3x + x = 3x + 1x = 4x There are pitfalls that are easy to fall into. Remember, you cannot combine unlike terms! Example Combine like terms: 3x 2 + x 3x 2 + x = 3x 2 + x (unlike terms can t be combined!) Simplifying Expressions There are two things (so far) to take care of when you are asked to simplify an expression: parentheses and like terms. So distribute to eliminate parentheses, then look for like terms that you can combine. Example Simplify: 4x x = 4x + 17 (the 8 and 9 were the only like terms) Example Simplify: 4(3m 2n) 4(3m 2n) = 4 3m 4 2n = 12m 8n (eliminate parens by distribution) Example Simplify: 6 + 3(4k + 5) 6 + 3(4k + 5) = k + 15 = 12k + 21 (distribute, then combine like terms) Example Simplify: 9 (7 6a) 9 (7 6a) = a = 2 + 6a (distribute the negative, then combine like terms)

32 32 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS 1. Evaluate x if x = 4. Skills Check 2. Is 2 a solution of 4y 7 = 2? 3. Simplify: 7x x 4. Simplify: 3y(x 2) 5. Simplify: 4(x 2) 3(2 x)

33 2.2. TRANSLATING ENGLISH PHRASES AND ALGEBRAIC EXPRESSIONS Translating English Phrases and Algebraic Expressions This section will be delayed until after we finish section 2.5 and then 3.4. Practice problems are available in the software: 2.2 Translating English Phrases and Algebraic Expressions Sometimes we are called upon to translate words in to a math problem. For instance, suppose Hercomer had 5 more than 12 bricks. Although in this case, it s easy to know that Hercomer had 17 bricks, later on we will get into more involved problems where you can t see the answer without doing some real math. Since it s hard to do math with words, we need to translate the words into what we call algebraic expressions Translating Words to Expressions using Variables In these phrases (which resemble English to some extent), there are certain code words used. We can substitute mathematical symbols for those words. Here are some hints: A number or some number means x (or you can choose a different letter) that number or the same number again refers to x another number or a different number means y, but not x sum means addition enclosed in parentheses difference means subtraction enclosed in parentheses product means multiplication enclosed in parentheses quotient means division enclosed in parentheses more than means addition less than means subtraction (with the order reversed) Examples 1. Translate A number subtracted from 48 Answer: 48 x (watch the order of the terms) 2. Translate The product of 6 and a number Answer: 6x (multiplication doesn t always have a symbol) 3. Translate 9 multiplied by the sum of a number and 5 Answer: 9(x + 5) (sum has implied parentheses) 4. Translate 7 less than a number Answer: x 7 (you must reverse the order to get the right answer)

34 34 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS Translating Algebraic Expressions into Words Sometimes we have to go the other way: from an algebraic expression into words. This is much less common, and generally it is best to stick with the simplest way you can say the words. If you try to get fancy, you have more chances to goof up. The following tables can help: Addition and Subtraction Word Op Comments Example Words Symbols plus + simple and preferred 2 plus sum + implied parentheses sum of 2 and 3 (2 + 3) minus simple and preferred 2 minus difference implied parentheses difference between 2 and 3 (2 3) Multiplication and Division Word Op Comments Example Words Symbols times simple and preferred 2 times product implied parentheses product of 2 and 3 (2 3) divided by simple and preferred 2 divided by 3 quotient implied parentheses quotient of 2 and or 3 2 (2 3) or ( ) 2 3 Now, after you check your skills, we can go back to where we came from on page 46. Skills Check 1. Write as an algebraic expression: Seven more than a number

35 2.3. SOLVING LINEAR EQUATIONS: X + B = C AND AX = C Solving Linear Equations: x + b = c and ax = c Practice problems are available in the software: 2.3a Solving Linear Equations Using Addition and Subtraction 2.3b Solving Linear Equations Using Multiplication and Division 2.3d Applications of Linear Equations: Multiplication and Division Addition Property of Equality An equation is like a two-pan balance with both sides weighted equally. If we add the same amount to both sides, they will still balance. We will use this principle to solve some linear equations. Procedure Undo what s with x Whatever is added to the variable x (or other variable), add the opposite to both sides. Example Solve: x 12 = 3 x 12 = x = 9 In this example, we added 12 to both sides because there was a -12 attached to the x. The opposite of -12 is 12. Adding the number -12 with its opposite 12 makes 0, so the left side becomes just plain x. We are done when x is all alone on one side of the equation. Example Solve: x + 9 = 12 x + 9 = 12 9 x = Solving Equations with the Variable on Both Sides We have a problem in getting the variable all by itself on one side of the equation when it shows up on both sides. We have to make it go away from one side or the other. While you can get the correct answer making either side give up its variable, most people don t want to work with negative numbers if they have a choice. Procedure Eliminate the little guy 1. Decide which term containing the variable is the little guy that will be eliminated. (a) For a case like 5x and 2x, 2x is the little guy. (b) For a case like 7x and 8x, 8x is the little guy.

36 36 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS (c) For a case like 3x and 2x, 3x is the little guy. 2. Add the opposite of the whole entire little guy term from both sides, combining like terms. Example Solve: 3x = 2x 2 2x 3x = 2x 2 x = 2 Notice a couple of things: We did not subtract x. We did not subtract 2. We subtracted the whole term 2x. We combined only the terms that are like terms. Don t try to force combining unlike terms. Example Solve: 7 2 m + 1 = 9 2 m 7 2 m simp 7 2 m + 1 = 2 9m 1 = 2 2 m 1 = m Simplifying an Equation Before Solving Sometimes we are given a big long mess of an equation to solve. Maybe the variable shows up five times. In such cases, we combine like terms before we solve. Watch out! Never combine like terms that are on opposite sides of the = symbol. The = symbol is like the Grand Canyon, and terms can t combine across the = symbol. They can t even see each other. Example Solve: 9r + 4r = 9r r 9r + 4r = 9r r like 13r + 4 = 12r r r + 4 = 4 4 r = 0 And sometimes we are given a complicated mess with parentheses in it. Those parentheses have to be eliminated. So we distribute to eliminate them, then continue on as above. Example Solve: 4(x + 1) (3x + 5) = 1 4(x + 1) (3x + 5) = 1 dist 4x + 4 3x 5 = 1 like x 1 = 1 +1 x = 2

37 2.3. SOLVING LINEAR EQUATIONS: X + B = C AND AX = C Solving Equations, Step by Step (First cut) To solve these linear equations in the best way, follow these steps in order: 1. If there are parentheses, distribute to get rid of them. (Watch your signs!) 2. Combine like terms, if any, on each side of the equation. (But not across the = symbol.) 3. If the variable appears on both sides of the equation, add the opposite of the whole little guy term to both sides. 4. Undo whatever is added to the variable by adding the opposite. 5. (Space Reserved for a method coming soon) Multiplication Property of Equality As mentioned before, an equation is like a two-pan balance with both sides weighted equally. But sometimes, adding the same amount to both sides does not help us get x all by itself. Consider 3x = 6. Adding 3 to both sides doesn t help. In fact, it makes a bigger mess, because the 3x and 3 are not like terms. We simply can t add to undo multiplication. However, simlar to the addition property from the previous section, we can multiply both sides by the same number, and they will still balance. And, since division is related to multiplication, we can also divide both sides by the same number. We will now use this principle to solve some more linear equations. Procedure Undo what s with x Whatever is multiplied times the variable x (or other variable), multiply by the reciprocal on both sides. Example Solve 3x = 6 3x = 6 13 x = 2 Remember that multiplying by 1 3 is the same as dividing by 3. So we can also use this technique with division. Procedure Alt. Undo what s with x Whatever is multiplied times the variable x (or other variable), divide both sides by that number. Example Solve 3x = 6 using the equivalent alternate method: 3x = 6 3 x = 2

38 38 CHAPTER 2. ALGEBRAIC EXPRESSIONS AND LINEAR EQUATIONS As you can see, the alternate division method is often the easiest. But when we have a fractional coefficient, the multiplication method works great. It is up to your personal preference which one to use. Example Solve 1 4 x = x = 6 4 x = Division Does Not Always Come Out Even Sometimes when we divide both sides of an equation by the coefficient of x, the plain number is not divisible by that coefficient. You calculator probably puts it into decimal form and rounds it off. This is bad form, unless specifically asked for in the instructions. Watch out! Rounded off values are generally wrong answers! The way we get around this is to give the answer in fractional form. Put the plain number upstairs, and the coefficient downstairs. Reduce if possible. Example Solve: 6x = 5 6x = 5 6 x = The Pesky x Sometimes we are solving an equation and get to the point where x = something. We don t want to know what x is. We want to know what x is. We can multiply both sides of the equation by 1 to finish solving. All that happens when we multiply everything by 1 is everything changes sign. Whatever was positive becomes negative, and whatever was negative becomes positive. They just flip-flop. You can also consider this dividing by 1. It amounts to the same thing. Example Solve x = 5 x = 5 ( 1) x = Simplifying Before Solving Sometimes we need to simplify an equation before we can solve it. We can combine like terms, as long as they are on the same side of the = sign, or distribute if there are parentheses. Our goal is always to get the variable in exactly one place, then to get the variable by itself. Example Solve: 4r 9r = 20 4r 9r = 20 like 5r = 20 ( 5) r = 4

39 2.3. SOLVING LINEAR EQUATIONS: X + B = C AND AX = C Solving Equations, Step by Step (Complete) Previously we had a step-by-step procedure for solving equations, but it had space reserved for another tool, namely multiplying both sides of the equation by the same number. Now that we have the last main tool for solving linear equations, here is the complete step-by-step list for how to solve linear equations. Follow these steps in order: 1. If there are parentheses, distribute to get rid of them. (Watch your signs!) 2. Combine like terms, if any, on each side of the equation. (But not across the = symbol.) 3. If the variable appears on both sides of the equation, subtract the whole little guy term from both sides. 4. Undo whatever is added to the variable by adding the opposite. 5. Undo whatever is multiplied onto the variable by dividing by that number (or multiplying by its reciprocal). 1. Solve: x 3 = 3 Skills Check 2. Solve: x + 3 = 3 3. Solve: 2x x 4 = 4(x 1) 4. Solve: 3x = 7 5. Solve: 4x + 2x = Solve: 3 5 x = 6