The Celsius temperature scale is based on the freezing point and the boiling point of water. 12 degrees Celsius below zero would be written as

Prealgebra, Chapter 2 - Integers, Introductory Algebra 2.1 Integers In the real world, numbers are used to represent real things, such as the height of a building, the cost of a car, the temperature of an oven, and many other things. In many cases, it makes sense to use a negative number. Example 1: Having $100 might be represented by the number Owing $100 might be represented by the number Example 2: The Celsius temperature scale is based on the freezing point and the boiling point of water. Water freezes: Water boils: And we can have temperatures that are below zero. 12 degrees Celsius below zero would be written as Example 3: A building is 70 feet tall. It also has a basement that goes 22 feet underground. The height of the building, written with a number: The depth of the basement, written with a number: Example 4: A car moving forward at 10 miles per hour. The speed is The car moves at the same speed in reverse. The speed is Example 5: Putting $150 into your bank account (a deposit): Taking $150 out of your bank account (a withdrawal): Integers and the number line Negative numbers to the left, positive to the right. The 0 position is referred to as the origin. The number 0 is neither positive nor negative. Positive and negative numbers are called. These numbers include all the numbers on the number line except 0. 2-1 Copyright 2011 by Lucid Education

The set of all integers consists of all of the counting numbers, their negatives, and 0. We can write this using set notation: Example: Plot the following numbers on the number line shown: -10, 1, 7, 13 Example: Circle the numbers below that are integers. 62 0.75 140 140.3-8 Ordering of Numbers All of the numbers are arranged in a particular order. Smaller numbers are toward the left and larger numbers are toward the right. Note that the farther a number is to the left, the smaller the number is. The number -5, therefore, is less than -3. Examples: -1000 1 80-99 Example: Write all of the following numbers in ascending order (smallest to largest). -1, 4, -16, 3, 11 Extreme Values If we have a set of numbers, such as { 4, -6, -2, 8, 1 } we call the least number the and we call the greatest number the Together, these are called the. 2-2 Copyright 2011 by Lucid Education

Example: For the set { 4, -6, -2, 8, 1 }, find the minimum and maximum values. Opposites and Absolute Value The negative of a number is sometimes called its opposite. The opposite of 7 is The opposite of -2 is The opposite of the opposite of 5 is The distance that a number is from the origin is called the of the number. You can visualize absolute value by looking at a number line. The absolute value of 5 is the distance that 5 is from 0. It is simply. The absolute value of -5 is the distance that -5 is from 0. This is also. The absolute value of a positive number is. The absolute value of a negative number is. The absolute value of 0 is. The absolute value of a number depends only on the number s distance from the origin, not on the sign of the number. Notation: Absolute value is indicated by two vertical bars. Examples: Simplify the following 2-3 Copyright 2011 by Lucid Education

2.1 Practice 1. Represent each of the following with a signed number. a) 500 ft above sea level. b) A loss of $96 c) An increase of 45 employees. d) 40 degrees below zero 2. Plot points on the number line for these numbers: 6, -12, 4, -7, 15 3. Which of the numbers shown are integers? Underline your answers 4 -½ 150-490 0.78 4. Place in ascending order. 4, -6, 8, 1, 2, -9, -10 5. Place in ascending order. 8, -6, -3, 0, 1, 4, -4, 5 6. Determine the maximum and minimum values. -5, -6, 2, 10, 11, 0, -9, -8 7. Determine the maximum and minimum values. 7, 15, 0, -18, 2, -5, -4, 3 8. Find the opposite of each number. a) 27 b) -3 c) -13 c) 16 9. Evaluate a) b) c) d) 2-5 Copyright 2011 by Lucid Education

10. Evaluate a) b) c) d) 11. Evaluate: 12. True or False. a) All integers are whole numbers. b) All integers can be written as fractions. c) All decimal numbers are integers. 13. Place absolute value bars to make the statement true. 14. Place absolute value bars to make the statement true. 15. For this list of numbers: 4, 1, 9, -8, -10 a) Which number is smallest? b) Which is farthest from the origin? c) Which has the largest absolute value? d) Which has the smallest absolute value? 16. Simplify the following. a) b) c) d) e) Describe the pattern that appears in parts a through d. 2-6 Copyright 2011 by Lucid Education

2.2 Adding Integers Now we will look again at addition, subtraction, multiplication, and division, but this time we will be using integers, which include negative as well as positive numbers. Whenever we do arithmetic, we have to pay attention to the sign of the numbers. On the number line, positive numbers are to the right and negative numbers are to the left. We can picture addition by using the number line. Adding a positive number is represented by moving to the right. Adding a negative number is represented by moving to the left. Example: 6 + 2 Example: (-6) + (-2) So we see that adding two positive numbers results in a larger positive number (farther to the right) and adding two negative numbers results in a larger absolute value negative number (farther to the left). When we add numbers with the same sign, we can simply add their absolute values, and then give the result the sign of the original numbers. Adding numbers with different signs Example: 7 + (-2) Note that the answer is positive. Example: (-6) + 4 Note that the answer is negative. When adding numbers with opposite signs, the sign of the answer is the same as the sign of the number with the largest absolute value. We can see this on a number line, but we don t want to draw a number line every time we need to add two numbers. Instead, we have a method that we can use that will allow us to add numbers quickly and easily. The method is this: To add two numbers with opposite sign, think of the absolute values of the numbers, and do subtract: larger minus smaller. Then give the result the same sign as the number with the larger absolute value. 2-7 Copyright 2011 by Lucid Education

Examples (do these mentally): 6 + (-11) (-8) + 12 (-2) + 19 (-17) + 5 One more simple but important point: The sum of any number and its opposite is 0. Example: 3 + (-3) = 0 This concept can be expressed as a general rule using a variable: This idea is known as the additive inverse property. Application Farmer Bob has a tree farm with 150 acres. There are 860 pine trees growing on the farm. If he cuts down 230 pines and then plants 300 new pines, how many pine trees does he then have growing on the farm? Practice Problems for section 2.2 2-8 Copyright 2011 by Lucid Education

2.2 Practice 1. 3. 3. 4. 5. 6. 7. 8. 9. Katerina has $489 in her bank account. She wrote a check for $23 and then placed $14 in her account. How much money does she have now? 10. Jake is playing video game soccer. He needs to go 100m. The distance traveled was 45m forward, 13 back, 6 forward, 8 back, 50 forward, and 14 back. How many yards does Jake have to go? 11. Katerina s book club originally had 14 girls. But then 8 left, 4 joined, 3 more joined, 5 departed, and 1 returned. How many people are now in Katerina s book club? 2-9 Copyright 2011 by Lucid Education

2.3 Subtracting Integers Subtracting a number is the same as. A subtraction problem can always be rewritten as an addition problem. Example: 7-2 can be written as This idea, as other ideas, can be expressed as a general rule by using variables: This is, in fact, the definition of subtraction. Subtracting a number means adding its opposite. Examples: Rewrite the following problems as addition problems and then solve the problems. 19-4 11-6 We can do this regardless of whether the numbers are positive or negative. Examples: -3-2 -20-12 8 - (-5) 17 - (-20) -6 - (-8) Application It is the year 2007. How many years has it been since the French Revolution in 1789? It is the year 2007. How many years has it been since Aristotle was born in 384 B.C.? 2-10 Copyright 2011 by Lucid Education

Subtraction on a Calculator Most calculators have two keys with a minus symbol on them. They are different keys and mean different things. One is a minus sign. It is used for subtraction. The other is a negative key, or a +/- key, which basically means negative or opposite. It is used to change the sign of a number. On some calculators it is the CHS key. CHS means change sign. Do the following on your calculator: 28 - (!6)!6 + 10 Practice Problems for section 2.3 HW 2B 2-11 Copyright 2011 by Lucid Education

2.3 Practice 1. 21 9 2. 56 34 3. 7 18 4. 10 52 5. 6 12 6. 24 13 7. 7 ( 13) 8. 24 ( 8) 9. 78 ( 90) 10. 12 ( 29) 11. Jake participated in 15 secret missions during 2011 but then he was sent on 3 fewer during the next year. How many total missions did he participate in during 2011 and 2012? 12. Katerina s bank says that her checking account has $568 in it. Katerina does not believe them. She then finds that there was a mistake was made in which $43 was recorded as a withdrawal when it should have been recorded as a deposit. What amount should she really have in her account? 13. Complete the statement: 6 ( 7) is equivalent to because. Solve the following on your calculator 14. 651 + ( 409) 15. 378 + ( 481) 16. 569 + ( 701) 17. 78 + ( 12) 18. 634 ( 56) 19. 185 (45) 2-12 Copyright 2011 by Lucid Education

2.4 Multiplying Integers We need to be able to multiply integers, both positive and negative. The rules are fairly simple: 1. Multiplying two positive numbers results in a. 2. Multiplying two negative numbers results in a. 3. Multiplying a positive and a negative results in a. According to the rules, solve the following: 5 C 3 = (-5)(3) = (5)(-3) = (-5)(-3) = These rules make sense if we think of the negative sign as meaning opposite. - 5 C 3 the opposite of five times three If we have two negative signs: (-5)(-3) the opposite of the opposite of five times three Think about the negative signs in these next examples: -(-5) = -(-(-5)) = -(-(-(-5))) = and so on An even number of negative signs will cancel out. An odd number of negative signs is always the same as a single negative sign. 2-13 Copyright 2011 by Lucid Education

This concept applies to multiplication. (-2)(-5)(-3) = (-8)(-5)(-2)(-3) = These ideas also allow us to deal with negative numbers raised to a power. Examples: (-5)(-5) = = The power of 2 means that we have two negative fives. We have an even number of negative signs. (-5)(-5)(-5) = = The power of 3 means that we have three negative fives. This is an odd number of negative signs. (-5)(-5)(-5)(-5) = = = = = Note that is not the same thing as means means If we don t have the parentheses, as in = 2-14 Copyright 2011 by Lucid Education

17. 18. 19. 20. 21. 22. 23. Imelda bought 76 pairs of shoes a year for 5 years in a row. How many pairs of shoes did she buy? 24. Jake s motorcycle is traveling at 80mph. He loses 1mph each minute for the next 15 minutes. How fast is he going at the end of th 15 minutes? 2-17 Copyright 2011 by Lucid Education

2.5 Dividing Integers When we divide, we handle the signs of the numbers in a manner similar to that used when multiplying. A positive number divided by a positive number results in a A positive number divided by a negative number results in a A negative number divided by a positive number results in a A negative number divided by a negative number results in a As before, we see that an even number of negative signs cancel each other out. We can state these same rules in a more concise manner: Dividing numbers with the same sign results in a positive number. Dividing numbers with opposite signs results in a negative number. Examples: = = = = Note that the following are all equivalent: The result is -5 in all three cases. In each of these fractions, we have a single negative sign which gets applied to our answer. Also, note that the following are all equivalent: In each of these cases, we have two negative signs. The negative signs end up canceling out, leaving us with a positive number for an answer. When a zero is involved, the same rules that apply to positive numbers also apply to negative numbers. Zero divided by any number is zero, and division by zero is undefined. = = 2-18 Copyright 2011 by Lucid Education

Division on a Calculator When dividing with negative numbers on a calculator, remember to use the negative key rather than the minus key to indicate a negative number. Do the following on your calculator: = = = Order of Operations Remember that the fraction bar acts as a grouping symbol. All the operations in the numerator must be performed, as well as all the operations in the denominator, before the division is done. Example: = Examples: = = = = = = Practice Problems for section 2.5 HW 2D 2-19 Copyright 2011 by Lucid Education

2.5 Practice Divide. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 16. Imelda spent $18 at each of 6 shoe stores. If she had $39 left, how much did she start with? 17. Jake and his three cousins went to the movies. The total cost was $73. How much did each person have to pay? 2-20 Copyright 2011 by Lucid Education

2.6 Variables and Expressions Variables When we are doing we use. Example: 2 + 3 When we are doing we use. Expressions Example: x + y Earlier we said that an expression was either a number by itself or multiple numbers combined with arithmetic operators ( +, -, x, ). Now we add that fact that expressions can also include variables. Expression: One or more numbers or variables combined with arithmetic operators. Using variables to write mathematical expressions We need to be able to take certain concepts, expressed in English, and write them mathematically. We will start with addition. The sum of a and b Five more than x d increased by 3" x plus y Four added to k Some things may look like expressions at first glance but are actually mathematically meaningless. Look at the following: 5 + x a + C b Things such as these are usually typographical errors. 2-21 Copyright 2011 by Lucid Education

Subtraction a minus b six less than x k decreased by 3" the difference of a and 20" Important point Addition is commutative: a + b = b + a Subtraction is not commutative: a - b b - a three less than a a less than 3" Multiplication Note that there are different ways to represent multiplication the product of five and x Simply writing 5x is usually considered preferable. three times n twice b the product of five, x, and y width times length Note that we typically use variable names that are meaningful. 2-22 Copyright 2011 by Lucid Education

Parentheses Parentheses are used to group numbers, variables, and operators. An expression in parentheses can be thought of as a single quantity. Examples: Five times the sum of x and y the product of w and the sum of a and b six times the difference of a and b the sum of a and b multiplied by the difference of a and b the product of h and two less than h three more than the product of 5 and t the sum of twice the width and twice the length Division In algebra, division is usually indicated with a fraction bar. a divided by 5" x divided by the sum of x and 5" the quotient of y and x the difference of L and y divided by the difference of c and x 2-23 Copyright 2011 by Lucid Education

Common Geometric Expressions Expressions that describe certain properties of geometric figures show up repeatedly in nearly all math classes. perimeter of a rectangle area of a rectangle volume of a box area of a triangle circumference of a circle area of a circle Practice Problems for section 2.6 HW 2E 2-24 Copyright 2011 by Lucid Education

2.6 Practice Write each of the following using variables and mathematical symbols. 1. The sum of a and b 2. m increased by 2 3. 5 more than n 4. 7 less than p 5. a decreased by 3 6. t minus 15 7. The product of b and c 8. The product of 12, p, and q 9. The product of 8 and the quantity x plus y 10. The sum of c and three times d 11. The product of x and 5 more than x 12. The quotient of m plus n, and 4 13. The difference w minus v divided by r 14. The sum of y and 7 divided by the difference of y and 7 Write each of the phrases, using the variable x to stand for the unnamed number. 15. 7 more than a number 2-25 Copyright 2011 by Lucid Education

16. A number decreased by 11 17. 15 decreased by a number 18. 6 times a number 19. A number divided by 10 20. 8 less than a number, divided by 5 Write each of the following using the variables given and the appropriate mathematical symbols. 21. π times the square of the radius 22. Twice the length (L) plus twice the width (W) 23. One half the product of the base (b) and the height (h) 24. Six times the length of one side (s) squared. 25. Identify which of the following are expressions. Underline the ones that are. 2-26 Copyright 2011 by Lucid Education

2.7 Evaluating Expressions If we have an algebraic expression, such as and we are given a value for each variable, such as a = 11, b = 7, c = 4 then we can find the value of the expression. We simply rewrite the expression, replacing each variable with the appropriate number. Simple expressions can be evaluated mentally or with a minimal amount of written work. Evaluate the following using the values a = 5, b = 2 2-27 Copyright 2011 by Lucid Education

When evaluating expressions, we need to pay attention to the order of operations. Evaluate the following, given that x = 2, y = 3 Pay careful attention to the order of operations. When evaluating expressions that contain a fraction bar, remember that the fraction bar groups the numerator and the denominator. We have to get a value for the numerator and a value for the denominator and then divide. Evaluate the following, given that x = 3, y = -1, z = 4 2-28 Copyright 2011 by Lucid Education

Evaluating expressions on a calculator Most calculators have parentheses keys. When evaluating an expression that contains a fraction bar, we have to remember to put parentheses around numerator and denominator. Example: Without the parentheses, the calculator would evaluate: 2 + 18 / 3-1 and the 18 divided by 3 would get done first, before the addition and subtraction. Squares and Negatives Be careful when a variable is squared. This can be tricky when the variable is negative or when it is subtracted. Evaluate the following for a = 4, b = -3 And remember that a fraction bar acts as a grouping symbol. 2-29 Copyright 2011 by Lucid Education

Sigma Notation The Greek letter Sigma, 3, in mathematics, stands for sum. Σx means the sum of x Find Σx for the given sets of numbers: 3, -8, 5, -2, 12 1, 2, 3, 4, 5, 6, 7, 8, 9 4, 8, -3, 7, 11, -5 Practice Problems for section 2.7 HW 2F 2-30 Copyright 2011 by Lucid Education

2.8 Simplifying Expressions Remember what an expression is: numbers and variables combined with arithmetic operators. Examples: An expression is made of terms that are added together. Note how many terms are in each of the following expressions: Remember that the terms are the parts of the expression that are added together. Each term can be a simple number, or a combination of numbers and variables multiplied together. Note that even though they look similar, the two term expression above is an entirely different thing than the single term expression. If a term is subtracted in the expression, the negative sign is part of the term. There are three terms in this expression. They are So each term has a sign, positive or negative. It is sometimes helpful to think of all the terms as being added together. This makes the sign of each term clear. 2-33 Copyright 2011 by Lucid Education

List the terms in each of the following expressions A term may have a numerical part and a variable part. The numerical part of a term is called the. If a term does not have a numerical part, such as we note that is the same thing as, so we recognize that the numerical part is 1. For the following expressions, list the coefficients of all of the terms in order. 2-34 Copyright 2011 by Lucid Education

Like Terms Like terms are terms. The variable parts must be exactly the same for the terms to be considered like terms. The following terms are all like terms: The following are not like terms because the variable parts are not the same. Even though the variables are all a and b, the variable parts are not exactly the same. In each of the following, find two terms that are like terms and circle them. Find three terms that are like terms in each of the following and circle them. Combining Like Terms All of the like terms in an expression can be combined into a single term. = Conceptually, this is the same as adding objects. + = 2 apples + 5 apples = 7 apples We can combine them into one number because they are the same thing. But we cannot combine apples and bananas because they are different types of things. + In the same way, we cannot combine terms that are not like terms. These terms cannot be combined. Terms have to be like terms to be combined. They must have exactly the same variable part. 2-35 Copyright 2011 by Lucid Education

In each of the following examples, simplify the expression by combining the like terms. Now, think about this example: + + 2 apples + 3 apples + 4 bananas Clearly, if we add these all together, we get 5 apples + 4 bananas + We combined the 2 apples and the 3 apples because they are the same think, apples, but we did not combine the apples and bananas. Next, instead of apples simply use the variable a, and instead of bananas use the variable b. 2a + 3a + 4b =. 2-36 Copyright 2011 by Lucid Education

Only like terms can be combined. In the following examples, decide which terms are like terms and then simplify the expression by combining like terms. Work carefully, and make sure that you understand every single one of these examples. 2-37 Copyright 2011 by Lucid Education

2.8 Practice 1. List the coefficients in each of these expressions. a) b) c) 2. Combine the like terms in the following expressions. a) b) c) d) e) f) g) h) i) j) k) 2-39 Copyright 2011 by Lucid Education

2.9 Introduction to Linear Equations What is an equation? An equation always has an equals sign. Examples: The equation is a common formula, the formula for the perimeter of a rectangle. The formula tells us how to find the distance around a rectangle if we know the length and the width. The equation is a true mathematical statement. It tells us that 2 plus 4 is equal to 6, and it is easy to see that this is, in fact, true. The equation, like the first equation, has a variable. We can t tell by looking whether or not the equation is true. It depends on the value of the variable x. In the equation the letter x is a variable. x could represent any number. But there is only one number that we can use for x that will make the equation true, the number 4. The number 4, in this case, is called the to the equation. In general, a solution to an equation is any value of the variable that will make the equation true. You can solve simple equations in your head if you can mentally add and subtract. x = a = Solving equations is one of the most important topics in algebra. We will learn more about solving equations as the course continues. 2-41 Copyright 2011 by Lucid Education

Verifying Solutions We can check a given number to see if it is a solution to an equation. To do this, simply substitute, or plug in the number in place of the variable. Example: Is 4 a solution to the equation? To solve this, we plug in the value 4 for the x in the equation, and then see if the resulting equation is a true statement. Example: Is 7 a solution to the equation? Example: Is 9 a solution to the equation? Example: Is 16 a solution to the equation? Equations with more than one solution Some equations have more than one solution. For example, There are two solutions: Note the exponent in the above equation. It is a 2. Because it is a power of 2, we call it a second degree equation. A second degree equation can have two solutions. The equation is a third degree equation. A third degree equation can have three solutions. In this course we will work mostly with first degree equations. First degree equations will have only one solution. 2-42 Copyright 2011 by Lucid Education

Examples: Identify each of the following as either an expression, a first degree equation, a second degree equation, or a third degree equation. Practice Problems for section 2.9 2-43 Copyright 2011 by Lucid Education

2.9 Practice 1. Determine if the number is a solution to the equation given. a) Is the number 5 a solution to? b) Is the number -4 a solution to? c) Is the number 7 a solution to? d) Is the number 50 a solution to? e) Is the number -3 a solution to? f) Is the number 4 a solution to 2. Label each of the following as an equation or an expression. a) b) c) d) e) 2-44 Copyright 2011 by Lucid Education

2.10 The Addition Property of Equality We will now learn an important technique that is used to solve equations. Engineers and scientists routinely techniques such as this in their everyday work. This technique is not difficult. It simply involves addition and subtraction. But it can be used to transform a very complicated equation into on that can be easily solved. An equation has two sides, a left side and a right side. The equation tells us that the expression on the left side is equal to the expression on the right side. Left side = Right side We can think of an equation as being similar to a seesaw. There is some weight on each side, and it is balanced at a point in the middle. In the picture above, a board is balanced on a point right at its center with some bricks on each side of the board. It will be perfectly balanced if there is the same weight on each side of the board. The left side must equal the right side. We can add some weights to the left side, but if we want to keep it balanced, we will have to add the same amount to the other side. In the same way, we can do things to an equation. We just have to keep the equation balanced. We have to make sure that whatever we do to one side of the equation we also do to the other side. Watch this example: This is a true statement. The left side does in fact equal the right side. We can take another number and add it to both sides: As long as we do the same thing to both sides, the equation is still a true statement, still mathematically correct. You should be able to see that this will always work. We can take any equation, and add anything to it, as long as we add the same thing to both sides. 2-45 Copyright 2011 by Lucid Education

This leads us to a very powerful and useful technique for solving equations. Suppose we have this problem: and we want to find x. That is, we want to solve the equation for x. Watch what happens when we add 18 to each side of the equation: This has helped us solve the equation. Now we know x, and finding x was our goal. The equation is solved. The +18 and the -18 on the left side canceled each other out. The variable x was left all by itself on one side of the equation. We say that x was. Solving an equation typically involves isolating the variable. Example: Add -7 to both sides. So we see that we can use this concept to solve equations. The concept is this: we can always add something, whatever we like, to one side of an equation, as long as we add the same thing to the other side. We can state this as a general principle: This principle is known as the addition property of equality. It works even if the equation is more complicated. Next we will look at a more complicated example. 2-46 Copyright 2011 by Lucid Education

Example: Example: Sometimes solving an equation involves more than one step. Example: Notice that these examples start with the variable appearing on both sides of the equation. Solving for x involves getting all the x s on one side of the equation. Choosing just the right thing to add or subtract helps us get rid of all the x s on one side, leaving the variable on the other side. Example: 2-47 Copyright 2011 by Lucid Education

The next example has a variable and a number of each side. Example: Sometimes the expression on one side or the other can be simplified. It is usually a good idea to simplify each side first before solving. Example: Example: Using the Distributive Property when Solving Equations Example: 2-48 Copyright 2011 by Lucid Education