Solving Equations by Adding and Subtracting

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Solving Equations by Adding and Subtracting"

Transcription

1 SECTION 2.1 Solving Equations by Adding and Subtracting 2.1 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the addition property to solve equations 3. Determine whether a given number is a solution for an application 4. Translate words to equation symbols 5. Solve application problems An equation such as x 3 5 is called a conditional equation because it can be either true or false depending on the value given to the variable. We could also use set-builder notation. We write {x x 2}, which is read, Every x such that x equals two. We will use both notations throughout the text. In this chapter you will begin working with one of the most important tools of mathematics, the equation. The ability to recognize and solve various types of equations is probably the most useful algebraic skill you will learn. We will continue to build upon the methods of this chapter throughout the remainder of the text. To start, let s describe what we mean by an equation. An equation is a mathematical statement that two expressions are equal. Some examples are 3 4 7, x 3 5, P 2L 2W. As you can see, an equals sign ( ) separates the two equal expressions. These expressions are usually called the left side and the right side of the equation. x 3 5 Left side Equals Right side An equation may be either true or false. For instance, is true because both sides name the same number. What about an equation such as x 3 5 that has a letter or variable on one side? Any number can replace x in the equation. However, only one number will make this equation a true statement is false If x is true is false } The number 2 is called the solution (or root) of the equation x 3 5 because substituting 2 for x gives a true statement. The set {2} is called the solution set. A solution for an equation is any value for the variable that makes the equation a true statement. 99

2 100 Chapter 2 Equations and Inequalities The solution set for an equation is the set of all values for the variables that make the equation a true statement. E x a m p l e 1 Verifying a Solution (a) Is 3 a solution for the equation 2x 4 10? To find out, replace x with 3 and evaluate 2x 4 on the left. Note that, until the left side equals the right side, a question mark is placed over the equals sign. Left side Right side Since is a true statement, 3 is a solution of the equation. The solution set is {3}. (b) Is 5 a solution of the equation 3x 2 2x 1? To find out, replace x with 5 and evaluate each side separately. Remember the rules for the order of operation. Multiply first; then add or subtract. Left side Right side Since the two sides do not name the same number, we do not have a true statement, and 5 is not a solution. CHECK YOURSELF 1 For the equation 2x 1 x 5 (a) Is 4 a solution? (b) Is 6 a solution?

3 Section 2.1 Solving Equations by Adding and Subtracting 101 You may be wondering whether an equation can have more than one solution. It certainly can. For instance, The equation x 2 9 is an example of a quadratic equation. We will consider methods of solution in Chapter 11. x 2 9 has two solutions. They are 3 and 3 because and ( 3) 2 9 The solution set is { 3, 3}. In this chapter, however, we will generally work with linear equations. These are equations that can be put into the form ax b 0 where the variable is x, where a and b are any numbers, and a is not equal to 0. In a linear equation, the variable can appear only to the first power. No other power (x 2, x 3, etc.) can appear. Linear equations are also called first-degree equations. The degree of an equation in one variable is the highest degree to which the variable appears. Linear equations in one variable that can be written in the form will have exactly one solution. ax b 0 a 0 E x a m p l e 2 Identifying Expressions and Equations Label each of the following as an expression, a linear equation, or an equation that is not linear. (a) 4x 5 is an expression. (b) 2x 8 0 is a linear equation. (c) 3x is not a linear equation. (d) 5x 15 is a linear equation. CHECK YOURSELF 2 Label each as an expression, a linear equation, or an equation that is not linear. (a) 2x 2 8 (b) 2x 3 0 (c) 5x 10 (d) 2x 1 7 One can find the solution for an equation such as x 3 8 by guessing the answer to the question What plus 3 is 8? Here the answer to the question is 5, and that is

4 102 Chapter 2 Equations and Inequalities also the solution for the equation. But for more complicated equations you are going to need something more than guesswork. A better method is to transform the given equation to an equivalent equation whose solution can be found by inspection. Let s make a definition. Equations that have the same solution are called equivalent equations. The following are all equivalent equations: 2x 3 5 2x 2 and x 1 They all have the same solution, 1. We say that a linear equation is solved when it is transformed to an equivalent equation of the form Note: In some cases we ll write the equation in the form x The number will be our solution when the equation has the variable isolated on the left or on the right. The variable is alone on the left side. x The right side is some number, the solution. The addition property of equality is the first property you will need to transform an equation to an equivalent form. Remember: An equation is a statement that the two sides are equal. Adding the same quantity to both sides does not change the equality or balance. The Addition Property of Equality If then a b a c b c In words, adding the same quantity to both sides of an equation gives an equivalent equation. Let s look at an example of applying this property to solve an equation. E x a m p l e 3 Using the Addition Property to Solve an Equation Solve x 3 9 Remember that our goal is to isolate x on one side of the equation. Since 3 is being subtracted from x, we can add 3 to remove it. We must use the addition property to add 3 to both sides of the equation.

5 Section 2.1 Solving Equations by Adding and Subtracting 103 To check, replace x with 12 in the original equation: x (True) Since we have a true statement, 12 is the solution. x 3 9 x 3 3 x 3 12 { Adding 3 undoes the subtraction and leaves x alone on the left. Since 12 is the solution for the equivalent equation x 12, it is the solution for our original equation. For either equation, the solution set is {12}. CHECK YOURSELF 3 Solve and check. x 5 4 The addition property also allows us to add a negative number to both sides of an equation. This is really the same as subtracting the same quantity from both sides. E x a m p l e 4 Using the Addition Property to Solve an Equation Solve x 5 9 Recall our comment that we could write an equation in the equivalent forms x or x, where represents some number. Suppose we have an equation like 12 x 7 Subtracting 7 will isolate x on the right: 12 x 7 17 x 7 15 x 7 and the solution is 5. In this case, 5 is added to x on the left. We can use the addition property to subtract 5 from both sides. This will undo the addition and leave the variable x alone on one side of the equation. x 5 9 x 5 5 x 5 4 The solution set is {4}. To check, replace x with 4: CHECK YOURSELF 4 Solve and check (True) x 6 13 What if the equation has a variable term on both sides? You will have to use the addition property to add or subtract a term involving the variable to get the desired result.

6 104 Chapter 2 Equations and Inequalities E x a m p l e 5 Using the Addition Property to Solve an Equation Solve 5x 4x 7 We will start by subtracting 4x from both sides of the equation. Do you see why? Remember that an equation is solved when we have an equivalent equation of the form x. 5x 4x 7 { 4x 4x 7 4x 4x 7 Subtracting 4x from both sides removes 4x from the right. To check: Since 7 is a solution for the equivalent equation x 7, it should be a solution for the original equation. To find out, replace x with 7: CHECK YOURSELF 5 Solve and check (True) 7x 6x 3 You may have to apply the addition property more than once to solve an equation. Look at Example 6. E x a m p l e 6 Using the Addition Property to Solve an Equation Solve 7x 8 6x We want all variables on one side of the equation. If we choose the left, we subtract 6x from both sides of the equation. This will remove 6x from the right: 7x 8 6x 6x 8 6x 6x 8 0x

7 Section 2.1 Solving Equations by Adding and Subtracting 105 We want the variable alone, so we add 8 to both sides. This isolates x on the left. x 8 0 x 8 8 x 8 8 The solution set is {8}. We ll leave it to you to check this result. CHECK YOURSELF 6 Solve and check. 9x 3 8x Often an equation will have more than one variable term and more than one number. You will have to apply the addition property twice in solving these equations. E x a m p l e 7 Using the Addition Property to Solve an Equation Solve 5x 7 4x 3 We would like the variable terms on the left, so we start by subtracting 4x to remove that term from the right side of the equation: 5x 7 4x 3 4x 7 4x 3 5x 7 4x 3 Now, to isolate the variable, we add 7 to both sides to undo the subtraction on the left: x 7 3 x 7 7 x 7 10 You could just as easily have added 7 to both sides and then subtracted 4x. The result would be the same. In fact, some students prefer to combine the two steps. The solution set is {10}. To check, replace x with 10 in the original equation: (True)

8 106 Chapter 2 Equations and Inequalities CHECK YOURSELF 7 Solve and check. (a) 4x 5 3x 2 (b) 6x 2 5x 4 Remember, by simplify we mean to combine all like terms. In solving an equation, you should always simplify each side as much as possible before using the addition property. E x a m p l e 8 Combining Like Terms and Solving the Equation Solve Like terms Like terms 5 8x 2 2x 3 5x Since like terms appear on each side of the equation, we start by combining the numbers on the left (5 and 2). Then we combine the like terms (2x and 5x) on the right. We have 3 8x 7x 3 Now we can apply the addition property, as before: 3 8x 7x 3 Subtract 7x. 3 7x 7x 3 Subtract 7x. 3 8x 7x 3 Subtract 7x 3 8x 7x 3 Subtract 3.x 3 8x 7x 6 Isolate x. The solution set is { 6}. To check, always return to the original equation. That will catch any possible errors in simplifying. Replacing x with 6 gives CHECK YOURSELF 8 Solve and check. 5 8( 6) 2 2( 6) 3 5( 6) (True) (a) 3 6x 4 8x 3 3x (b) 5x 21 3x 20 7x 2

9 Section 2.1 Solving Equations by Adding and Subtracting 107 We may have to apply some of the properties discussed in Section 0.4 in solving equations. Example 9 illustrates the use of the distributive property to clear an equation of parentheses. E x a m p l e 9 Note: 2(3x 4) 2(3x) 2(4) 6x 8 Using the Distributive Property and Solving Equations Solve 2(3x 4) 5x 6 Applying the distributive property on the left, we have 6x 8 5x 6 We can then proceed as before: Remember that x 14 and 14 x are equivalent equations. 6x 8 5x 6 Subtract 5x. 5x 8 5x 6 Subtract 5x. 5x 8 5x 6 Subtract 5x. 5x 8 5x 8 Subtract 8.X 5x 8 5x 14 Subtract 5x. The solution set is { 14}. We will leave the checking of this result to the reader. Remember: Always return to the original equation to check. CHECK YOURSELF 9 Solve and check each of the following equations. (a) 4(5x 2) 19x 4 (b) 3(5x 1) 2(7x 3) 4 The main reason for learning how to set up and solve algebraic equations is so that we can use them to solve word problems. In fact, algebraic equations were invented to make solving word problems much easier. The first word problems that we know about are over 4000 years old. They were literally written in stone, on Babylonian tablets, about 500 years before the first algebraic equation made its appearance. Before algebra, people solved word problems primarily by substitution, which is a method of finding unknown integers by using trial and error in a logical way. Example 10 shows how to solve a word problem using substitution.

10 108 Chapter 2 Equations and Inequalities E x a m p l e 1 0 Solving a Word Problem by Substitution The sum of two consecutive integers is 37. Find the two integers. If the two integers were 20 and 21, their sum would be 41. Since that s more than 37, the integers must be smaller. If the integers were 15 and 16, the sum would be 31. More trials yield that the sum of 18 and 19 is 37. CHECK YOURSELF 10 The sum of two consecutive integers is 91. Find the two integers. Most word problems are not so easily solved by substitution. For more complicated word problems, a five-step procedure is used. Using this step-by-step approach will, with practice, allow you to organize your work. Organization is the key to solving word problems. Here are the five steps. To Solve Word Problems Step 1 Step 2 Step 3 Step 4 Step 5 Read the problem carefully. Then reread it to decide what you are asked to find. Choose a letter to represent one of the unknowns in the problem. Then represent all other unknowns of the problem with expressions that use the same letter. Translate the problem to the language of algebra to form an equation. Solve the equation and answer the question of the original problem. Check your solution by returning to the original problem. We discussed these translations in Section 1.1. You might find it helpful to review that section before going on. The third step is usually the hardest part. We must translate words to the language of algebra. Before we look at a complete example, the following table may help you review that translation step.

11 Section 2.1 Solving Equations by Adding and Subtracting 109 Translating Words to Algebra Words Algebra The sum of x and y x y 3 plus a 3 a or a 3 5 more than m m 5 b increased by 7 b 7 The difference of x and y x y 4 less than a a 4 s decreased by 8 s 8 The product of x and y x y or xy 5 times a 5 a or 5a Twice m 2m The quotient of x and y x y a divided by 6 One-half of b 2 b or 1 2 b 6 a Now let s look at some typical examples of translating phrases to algebra. E x a m p l e 1 1 Translating Statements Translate each statement to an algebraic expression. (a) the sum of a and 2 times b a 2b Sum 2 times b (b) 5 times m increased by 1 5m 1 5 times m Increased by 1 (c) 5 less than 3 times x 3x 5 3 times x 5 less than (d) The product of x and y, divided by 3 x y 3 The product of x and y Divided by 3

12 110 Chapter 2 Equations and Inequalities CHECK YOURSELF 11 Translate to algebra. (a) 2 more than twice x (b) 4 less than 5 times n (c) The product of twice a and b (d) The sum of s and t, divided by 5 Now let s work through a complete example. Although this problem could be solved by substitution, it is presented here to help you practice the five-step approach. E x a m p l e 1 2 Solving an Application The sum of a number and 5 is 17. What is the number? Step 1 Step 2 Step 3 Read carefully. You must find the unknown number. Choose letters or variables. Let x represent the unknown number. There are no other unknowns. Translate. Step 4 Solve. The sum of x 5 17 is Always return to the original problem to check your result and not to the equation of step 3. This will prevent possible errors! x 5 17 x Subtract 5. x 12 So the number is 12. Step 5 Check. Is the sum of 12 and 5 equal to 17? Yes ( ). We have checked our solution.

13 Section 2.1 Solving Equations by Adding and Subtracting 111 CHECK YOURSELF 12 The sum of a number and 8 is 35. What is the number? CHECK YOURSELF ANSWERS 1. (a) 4 is not a solution; (b) 6 is a solution. 2. (a) Nonlinear equation; (b) linear equation; (c) expression; (d) linear equation. 3. {9}. 4. {7}. 5. {3}. 6. { 3}. 7. (a) {7}; (b) { 6}. 8. (a) { 10}; (b) { 3}. 9. (a) {12}; (b) { 13}. 10. {45, 46}. 11. (a) 2x 2; (b) 5n 4; (c) 2ab; (d) s t The equation is x The number is 27.

14 E x e r c i s e s Yes 2. No 3. No 4. Yes 5. No 6. Yes 7. Yes 8. No 9. No 10. Yes 11. No 12. Yes 13. Yes 14. No 15. Yes 16. No 17. Yes 18. No 19. No 20. Yes 21. Yes 22. No 23. Linear equation 24. Expression 25. Expression 26. Linear equation 27. Linear equation 28. Not a linear equation 29. {2} 30. {10} 31. {11} 32. {4} 33. { 2} 34. { 3} 35. {6} 36. { 7} 37. {4} 38. { 8} Is the number shown in parentheses a solution for the given equation? 1. x 4 9 (5) 2. x 2 11 (8) 3. x 15 6 ( 21) 4. x 11 5 (16) 5. 5 x 2 (4) x 7 (3) 7. 4 x 6 ( 2) 8. 5 x 6 ( 3) 9. 3x 4 13 (8) 10. 5x 6 31 (5) 11. 4x 5 7 (2) 12. 2x 5 1 (3) x 7 ( 1) x 9 ( 2) 15. 4x 5 2x 3 (4) 16. 5x 4 2x 10 (4) 17. x 3 2x 5 x 8 (5) 18. 5x 3 2x 3 x 12 ( 2) x 18 (20) x 24 (40) x 5 11 (10) x 8 12 ( 6) 3 Label each of the following as an expression, or a linear equation, or an equation that is not linear x x x 2x x x x 2 5x 2 5 Solve each equation and check your results. Express each answer in set notation. 29. x x x x x x x x x 3x x 6x 8

15 Section 2.1 Solving Equations by Adding and Subtracting { 10} 40. {5} 41. { 3} 42. {6} 43. {2} 44. {3} 45. {4} 46. { 7} 47. {6} 48. { 3} 49. {6} 50. {11} 51. {6} 52. { 7} 53. { 18} 54. {13} 55. {16} 56. { 17} 57. {8} 58. { 11} 59. {2} 60. {3} 61. x x x 7 2x 64. 5x 4 6x 65. 2(x 5) x (x 7) 4x 67. c 68. d 69. a 70. d x 10x x 8x x 3 5x x 6 11x 43. 2x 3 x x 2 2x x 7 4x x 5 7x x 2 6x x 3 9x x 2 3x 11 2x 50. 6x 3 2x 7x x 7 3x 5x 13 x 52. 5x 9 4x 9 8x (3x 4) 11x (5x 3) 9x (7x 2) 5(4x 1) (5x 3) 3(8x 2) x x x 3 2 x x x x x In Exercises 61 to 66, translate each statement to an algebraic equation. Let x represent the number in each case more than a number is less than a number is less than 3 times a number is twice that same number more than 5 times a number is 6 times that same number times the sum of a number and 5 is 18 more than that same number times the sum of a number and 7 is 4 times that same number. 67. Which of the following is equivalent to the equation 8x 5 9x 4? (a) 17x 9 (b) x 9 (c) 8x 9 9x (d) 9 17x 68. Which of the following is equivalent to the equation 5x 7 4x 12? (a) 9x 19 (b) 9x 7 12 (c) x 18 (d) x Which of the following is equivalent to the equation 12x 6 8x 14? (a) 4x 6 14 (b) x 20 (c) 20x 20 (d) 4x Which of the following is equivalent to the equation 7x 5 12x 10? (a) 5x 15 (b) 7x 5 12x (c) 5 5x (d) 7x 15 12x

16 114 Chapter 2 Equations and Inequalities 71. True 72. False ; x ; x ; x ; x ; 1840 x $1360; x $290; x $1325; x $740; x True or false? 71. Every linear equation with one variable has exactly one solution. 72. Isolating the variable on the right side of the equation will result in a negative solution. Solve the following word problems. Be sure to label the unknowns and to show the equation you use for the solution. 73. Number problem. The sum of a number and 7 is 33. What is the number? 74. Number problem. The sum of a number and 15 is 22. What is the number? 75. Number problem. The sum of a number and 15 is 7. What is the number? 76. Number problem. The sum of a number and 8 is 17. What is the number? 77. Number of votes cast. In an election, the winning candidate has 1840 votes. If the total number of votes cast was 3260, how many votes did the losing candidate receive? 78. Monthly earnings. Mike and Stefanie work at the same company and make a total of $2760 per month. If Stefanie makes $1400 per month, how much does Mike earn every month? 79. Appliance costs. A washer-dryer combination costs $650. If the washer costs $360, what does the dryer cost? 80. Computer costs. You have $2350 saved for the purchase of a new computer that costs $3675. How much more must you save? 81. Price increases. The price of an item has increased by $225 over last year. If the item is now selling for $965, what was the price last year? 82. An algebraic equation is a complete sentence. It has a subject, a verb, and a predicate. For example, x 2 5 can be written in English as Two more than a number is five. Or, A number added to two is five. Write an English version of the following equations. Be sure you write complete sentences and that the sentences express the same idea as the equations. Exchange sentences with another student, and see if your interpretation of each other s sentences result in the same equation. (a) 2x 5 x 1 (b) 2(x 2) 14 (c) n 5 n 6 2 (d) 7 3a 5 a 83. Complete the following explanation in your own words: The difference between 3(x 1) 4 2x and 3(x 1) 4 2x is I make $2.50 an hour more in my new job. If x the amount I used to make per hour and y the amount I now make, which equation(s) below say the same thing as the statement above? Explain your choice(s) by translating the equation into English and comparing with the original statement. (a) x y 2.50 (b) x y 2.50 (c) x 2.50 y (d) 2.50 y x (e) y x 2.50 (f) 2.50 x y

17 Section 2.1 Solving Equations by Adding and Subtracting The river rose 4 feet above flood stage last night. If a the river s height at flood stage and b the river s height now (the morning after), which equations below say the same thing as the statement? Explain your choices by translating the equations into English and comparing the meaning with the original statement. (a) a b 4 (b) b 4 a (c) a 4 b (d) a 4 b (e) b 4 b (f) b a Surprising Results! Work with other students to try this experiment. Each person should do the following six steps mentally, not telling anyone else what their calculations are: (a) Think of a number. (b) Add 7. (c) Multiply by 3. (d) Add 3 more than the original number. (e) Divide by 4. (f) Subtract the original number. What number do you end up with? Compare your answer with everyone else s. Does everyone have the same answer? Make sure that everyone followed the directions accurately. How do you explain the results? Algebra makes the explanation clear. Work together to do the problem again, using a variable for the number. Make up another series of computations that give surprising results. 87. (a) Do you think that the following is a linear equation in one variable? 3(2x 5) 6(x 2) (b) What happens when you use the properties of this section to solve the equation? (c) Pick any number to substitute for x in this equation. Now try a different number to substitute for x in the equation. Try yet another number to substitute for x in the equation. Summarize your findings. (d) Can this equation be called linear in one variable? Refer to the definition as you explain your answer. 88. (a) Do you think the following is a linear equation in one variable? 4(3x 5) 2(6x 8) 3 (b) What happens when you use the properties of this section to solve the equation? (c) Do you think it is possible to find a solution for this equation? (d) Can this equation be called linear in one variable? Refer to the definition as you explain your answer.

Section 2.1 Objective 1: Determine If a Number Is a Solution of an Equation Video Length 5:19. Definition A in is an equation that can be

Section 2.1 Objective 1: Determine If a Number Is a Solution of an Equation Video Length 5:19. Definition A in is an equation that can be Section 2.1 Video Guide Linear Equations: The Addition and Multiplication Properties of Equality Objectives: 1. Determine If a Number Is a Solution of an Equation 2. Use the Addition Property of Equality

More information

33. SOLVING LINEAR INEQUALITIES IN ONE VARIABLE

33. SOLVING LINEAR INEQUALITIES IN ONE VARIABLE get the complete book: http://wwwonemathematicalcatorg/getfulltextfullbookhtm 33 SOLVING LINEAR INEQUALITIES IN ONE VARIABLE linear inequalities in one variable DEFINITION linear inequality in one variable

More information

Adding and Subtracting Terms

Adding and Subtracting Terms Adding and Subtracting Terms 1.6 OBJECTIVES 1.6 1. Identify terms and like terms 2. Combine like terms 3. Add algebraic expressions 4. Subtract algebraic expressions To find the perimeter of (or the distance

More information

1.4 Mathematical Equivalence

1.4 Mathematical Equivalence 1.4 Mathematical Equivalence Introduction a motivating example sentences that always have the same truth values can be used interchangeably the implied domain of a sentence In this section, the idea of

More information

For all For every For each For any There exists at least one There exists There is Some

For all For every For each For any There exists at least one There exists There is Some Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following

More information

GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using

GRADE 7 MATH LEARNING GUIDE. Lesson 26: Solving Linear Equations and Inequalities in One Variable Using GRADE 7 MATH LEARNING GUIDE Lesson 26: Solving Linear Equations and Inequalities in One Variable Using Guess and Check Time: 1 hour Prerequisite Concepts: Evaluation of algebraic expressions given values

More information

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models

Mini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic expressions.. Translate English phrases into algebraic expressions.. Determine whether a number is a solution

More information

Exponents. Reteach. Write each expression in exponential form (0.4)

Exponents. Reteach. Write each expression in exponential form (0.4) 9-1 Exponents You can write a number in exponential form to show repeated multiplication. A number written in exponential form has a base and an exponent. The exponent tells you how many times a number,

More information

9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)

9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality) 9 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same

More information

Chapter 1. ANALYZE AND SOLVE LINEAR EQUATIONS (3 weeks)

Chapter 1. ANALYZE AND SOLVE LINEAR EQUATIONS (3 weeks) Chapter 1. ANALYZE AND SOLVE LINEAR EQUATIONS (3 weeks) Solve linear equations in one variable. 8EE7ab In this Chapter we review and complete the 7th grade study of elementary equations and their solution

More information

30. TRANSFORMING TOOL #1 (the Addition Property of Equality)

30. TRANSFORMING TOOL #1 (the Addition Property of Equality) 30 TRANSFORMING TOOL #1 (the Addition Property of Equality) sentences that look different, but always have the same truth values What can you DO to a sentence that will make it LOOK different, but not

More information

8. TRANSFORMING TOOL #1 (the Addition Property of Equality)

8. TRANSFORMING TOOL #1 (the Addition Property of Equality) 8 TRANSFORMING TOOL #1 (the Addition Property of Equality) sentences that look different, but always have the same truth values What can you DO to a sentence that will make it LOOK different, but not change

More information

25. REVISITING EXPONENTS

25. REVISITING EXPONENTS 25. REVISITING EXPONENTS exploring expressions like ( x) 2, ( x) 3, x 2, and x 3 rewriting ( x) n for even powers n This section explores expressions like ( x) 2, ( x) 3, x 2, and x 3. The ideas have been

More information

STUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition

STUDY GUIDE Math 20. To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition STUDY GUIDE Math 0 To the students: To accompany Intermediate Algebra for College Students By Robert Blitzer, Third Edition When you study Algebra, the material is presented to you in a logical sequence.

More information

Chapter 1 Review of Equations and Inequalities

Chapter 1 Review of Equations and Inequalities Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve

More information

Prealgebra. Edition 5

Prealgebra. Edition 5 Prealgebra Edition 5 Prealgebra, Edition 5 2009, 2007, 2005, 2004, 2003 Michelle A. Wyatt (M A Wyatt) 2009, Edition 5 Michelle A. Wyatt, author Special thanks to Garry Knight for many suggestions for the

More information

Lesson 6: Algebra. Chapter 2, Video 1: "Variables"

Lesson 6: Algebra. Chapter 2, Video 1: Variables Lesson 6: Algebra Chapter 2, Video 1: "Variables" Algebra 1, variables. In math, when the value of a number isn't known, a letter is used to represent the unknown number. This letter is called a variable.

More information

1.1 The Language of Mathematics Expressions versus Sentences

1.1 The Language of Mathematics Expressions versus Sentences The Language of Mathematics Expressions versus Sentences a hypothetical situation the importance of language Study Strategies for Students of Mathematics characteristics of the language of mathematics

More information

WRITING EQUATIONS 4.1.1

WRITING EQUATIONS 4.1.1 WRITING EQUATIONS 4.1.1 In this lesson, students translate written information, often modeling everyday situations, into algebraic symbols and linear equations. Students use let statements to specifically

More information

1.2 The Role of Variables

1.2 The Role of Variables 1.2 The Role of Variables variables sentences come in several flavors true false conditional In this section, a name is given to mathematical sentences that are sometimes true, sometimes false they are

More information

Prepared by Sa diyya Hendrickson. Package Summary

Prepared by Sa diyya Hendrickson. Package Summary Introduction Prepared by Sa diyya Hendrickson Name: Date: Package Summary Understanding Variables Translations The Distributive Property Expanding Expressions Collecting Like Terms Solving Linear Equations

More information

WRITING EQUATIONS through 6.1.3

WRITING EQUATIONS through 6.1.3 WRITING EQUATIONS 6.1.1 through 6.1.3 An equation is a mathematical sentence that conveys information to the reader. It uses variables and operation symbols (like +, -, /, =) to represent relationships

More information

PLEASE NOTE THAT YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

PLEASE NOTE THAT YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR! DETAILED SOLUTIONS AND CONCEPTS - INTRODUCTION TO ALGEBRA Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

More information

Chapter Three: Translations & Word Problems

Chapter Three: Translations & Word Problems Chapter Three: Translations & Word Problems Index: A: Literal Equations B: Algebraic Translations C: Consecutive Word Problems D: Linear Word Problems Name: Date: Period: Algebra I Literal Equations 3A

More information

Chapter 5 Simplifying Formulas and Solving Equations

Chapter 5 Simplifying Formulas and Solving Equations Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify

More information

Writing and Graphing Inequalities

Writing and Graphing Inequalities .1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle

More information

Reteach Simplifying Algebraic Expressions

Reteach Simplifying Algebraic Expressions 1-4 Simplifying Algebraic Expressions To evaluate an algebraic expression you substitute numbers for variables. Then follow the order of operations. Here is a sentence that can help you remember the order

More information

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2

NAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2 5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate

More information

Math 90 Lecture Notes Chapter 1

Math 90 Lecture Notes Chapter 1 Math 90 Lecture Notes Chapter 1 Section 1.1: Introduction to Algebra This textbook stresses Problem Solving! Solving problems is one of the main goals of mathematics. Think of mathematics as a language,

More information

31. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)

31. TRANSFORMING TOOL #2 (the Multiplication Property of Equality) 3 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same

More information

Algebra Year 10. Language

Algebra Year 10. Language Algebra Year 10 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.

More information

('')''* = 1- $302. It is common to include parentheses around negative numbers when they appear after an operation symbol.

('')''* = 1- $302. It is common to include parentheses around negative numbers when they appear after an operation symbol. 2.2 ADDING INTEGERS Adding Integers with the Same Sign We often associate the + and - symbols with positive and negative situations. We can find the sum of integers by considering the outcome of these

More information

Systems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1

Systems of Equations. Red Company. Blue Company. cost. 30 minutes. Copyright 2003 Hanlonmath 1 Chapter 6 Systems of Equations Sec. 1 Systems of Equations How many times have you watched a commercial on television touting a product or services as not only the best, but the cheapest? Let s say you

More information

irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways:

irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways: CH 2 VARIABLES INTRODUCTION F irst we need to know that there are many ways to indicate multiplication; for example the product of 5 and 7 can be written in a variety of ways: 5 7 5 7 5(7) (5)7 (5)(7)

More information

2 Introduction to Variables

2 Introduction to Variables www.ck12.org CHAPTER 2 Introduction to Variables Chapter Outline 2.1 VARIABLE EXPRESSIONS 2.2 PATTERNS AND EXPRESSIONS 2.3 COMBINING LIKE TERMS 2.4 THE DISTRIBUTIVE PROPERTY 2.5 ADDITION AND SUBTRACTION

More information

Mini Lecture 2.1 The Addition Property of Equality

Mini Lecture 2.1 The Addition Property of Equality Mini Lecture.1 The Addition Property of Equality 1. Identify linear equations in one variable.. Use the addition property of equality to solve equations.. Solve applied problems using formulas. 1. Identify

More information

Conceptual Explanations: Simultaneous Equations Distance, rate, and time

Conceptual Explanations: Simultaneous Equations Distance, rate, and time Conceptual Explanations: Simultaneous Equations Distance, rate, and time If you travel 30 miles per hour for 4 hours, how far do you go? A little common sense will tell you that the answer is 120 miles.

More information

1.1 Variables and Expressions How can a verbal expression be translated to an algebraic expression?

1.1 Variables and Expressions How can a verbal expression be translated to an algebraic expression? 1.1 Variables and Expressions How can a verbal expression be translated to an algebraic expression? Recall: Variable: Algebraic Expression: Examples of Algebraic Expressions: Different ways to show multiplication:

More information

32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE

32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE get the complete book: /getfulltextfullbook.htm 32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE classifying families of sentences In mathematics, it is common to group together sentences of the same type

More information

Math 1 Variable Manipulation Part 1 Algebraic Equations

Math 1 Variable Manipulation Part 1 Algebraic Equations Math 1 Variable Manipulation Part 1 Algebraic Equations 1 PRE ALGEBRA REVIEW OF INTEGERS (NEGATIVE NUMBERS) Numbers can be positive (+) or negative (-). If a number has no sign it usually means that it

More information

Section 2.5 Linear Inequalities

Section 2.5 Linear Inequalities Section 2.5 Linear Inequalities WORDS OF COMPARISON Recently, you worked with applications (word problems) in which you were required to write and solve an equation. Sometimes you needed to translate sentences

More information

MATH 60 Course Notebook Chapter #1

MATH 60 Course Notebook Chapter #1 MATH 60 Course Notebook Chapter #1 Integers and Real Numbers Before we start the journey into Algebra, we need to understand more about the numbers and number concepts, which form the foundation of Algebra.

More information

Polynomials. This booklet belongs to: Period

Polynomials. This booklet belongs to: Period HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher

More information

Essentials of Intermediate Algebra

Essentials of Intermediate Algebra Essentials of Intermediate Algebra BY Tom K. Kim, Ph.D. Peninsula College, WA Randy Anderson, M.S. Peninsula College, WA 9/24/2012 Contents 1 Review 1 2 Rules of Exponents 2 2.1 Multiplying Two Exponentials

More information

Word Problems. Mathematics Division, IMSP, UPLB

Word Problems. Mathematics Division, IMSP, UPLB Word Problems Objectives Upon completion, you should be able to: Translate English statements into mathematical statements Use the techniques learned in solving linear, quadratic and systems of equations

More information

Algebra Terminology Part 1

Algebra Terminology Part 1 Grade 8 1 Algebra Terminology Part 1 Constant term or constant Variable Numerical coefficient Algebraic term Like terms/unlike Terms Algebraic expression Algebraic equation Simplifying Solving TRANSLATION

More information

3.5 Solving Equations Involving Integers II

3.5 Solving Equations Involving Integers II 208 CHAPTER 3. THE FUNDAMENTALS OF ALGEBRA 3.5 Solving Equations Involving Integers II We return to solving equations involving integers, only this time the equations will be a bit more advanced, requiring

More information

3.0 Distributive Property and Expressions Teacher Notes

3.0 Distributive Property and Expressions Teacher Notes 3.0 Distributive Property and Expressions Teacher Notes Distributive Property: To multiply a sum or difference by a number, multiply each number in the sum or difference by the number outside of the parentheses.

More information

ALGEBRA CLAST MATHEMATICS COMPETENCIES

ALGEBRA CLAST MATHEMATICS COMPETENCIES 2 ALGEBRA CLAST MATHEMATICS COMPETENCIES IC1a: IClb: IC2: IC3: IC4a: IC4b: IC: IC6: IC7: IC8: IC9: IIC1: IIC2: IIC3: IIC4: IIIC2: IVC1: IVC2: Add and subtract real numbers Multiply and divide real numbers

More information

we first add 7 and then either divide by x - 7 = 1 Adding 7 to both sides 3 x = x = x = 3 # 8 1 # x = 3 # 4 # 2 x = 6 1 =?

we first add 7 and then either divide by x - 7 = 1 Adding 7 to both sides 3 x = x = x = 3 # 8 1 # x = 3 # 4 # 2 x = 6 1 =? . Using the Principles Together Applying Both Principles a Combining Like Terms a Clearing Fractions and Decimals a Contradictions and Identities EXAMPLE Solve: An important strategy for solving new problems

More information

1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS

1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS 1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS Combining Like Terms In algebra we often deal with terms such as 4y or 7x. What do we mean by terms? A term is a number, a variable, or a product of a number

More information

Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2

Name Period Date. polynomials of the form x ± bx ± c. Use guess and check and logic to factor polynomials of the form 2 Name Period Date POLYNOMIALS Student Packet 3: Factoring Polynomials POLY3 STUDENT PAGES POLY3.1 An Introduction to Factoring Polynomials Understand what it means to factor a polynomial Factor polynomials

More information

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities

Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,

More information

Polynomials; Add/Subtract

Polynomials; Add/Subtract Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x

More information

27. THESE SENTENCES CERTAINLY LOOK DIFFERENT

27. THESE SENTENCES CERTAINLY LOOK DIFFERENT 27 HESE SENENCES CERAINLY LOOK DIEREN comparing expressions versus comparing sentences a motivating example: sentences that LOOK different; but, in a very important way, are the same Whereas the = sign

More information

Week 7 Algebra 1 Assignment:

Week 7 Algebra 1 Assignment: Week 7 Algebra 1 Assignment: Day 1: Chapter 3 test Day 2: pp. 132-133 #1-41 odd Day 3: pp. 138-139 #2-20 even, 22-26 Day 4: pp. 141-142 #1-21 odd, 25-30 Day 5: pp. 145-147 #1-25 odd, 33-37 Notes on Assignment:

More information

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology

Intermediate Algebra. Gregg Waterman Oregon Institute of Technology Intermediate Algebra Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International license. The essence of the license

More information

ALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations

ALGEBRA 1. Interactive Notebook Chapter 2: Linear Equations ALGEBRA 1 Interactive Notebook Chapter 2: Linear Equations 1 TO WRITE AN EQUATION: 1. Identify the unknown (the variable which you are looking to find) 2. Write the sentence as an equation 3. Look for

More information

Section 2.2 Objectives

Section 2.2 Objectives Section 2.2 Objectives Solve multi-step equations using algebra properties of equality. Solve equations that have no solution and equations that have infinitely many solutions. Solve equations with rational

More information

Lesson 8: Using If-Then Moves in Solving Equations

Lesson 8: Using If-Then Moves in Solving Equations Student Outcomes Students understand and use the addition, subtraction, multiplication, division, and substitution properties of equality to solve word problems leading to equations of the form and where,,

More information

Chapter 9: Roots and Irrational Numbers

Chapter 9: Roots and Irrational Numbers Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic

More information

Lesson 24: True and False Number Sentences

Lesson 24: True and False Number Sentences NYS COMMON CE MATHEMATICS CURRICULUM Lesson 24 6 4 Student Outcomes Students identify values for the variables in equations and inequalities that result in true number sentences. Students identify values

More information

27. THESE SENTENCES CERTAINLY LOOK DIFFERENT

27. THESE SENTENCES CERTAINLY LOOK DIFFERENT get the complete book: http://wwwonemathematicalcatorg/getullextullbookhtm 27 HESE SENENCES CERAINLY LOOK DIEREN comparing expressions versus comparing sentences a motivating example: sentences that LOOK

More information

Direct Proofs. the product of two consecutive integers plus the larger of the two integers

Direct Proofs. the product of two consecutive integers plus the larger of the two integers Direct Proofs A direct proof uses the facts of mathematics and the rules of inference to draw a conclusion. Since direct proofs often start with premises (given information that goes beyond the facts of

More information

ALGEBRA 1 SUMMER ASSIGNMENT

ALGEBRA 1 SUMMER ASSIGNMENT Pablo Muñoz Superintendent of Schools Amira Presto Mathematics Instructional Chair Summer Math Assignment: The Passaic High School Mathematics Department requests all students to complete the summer assignment.

More information

Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero

Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

Equations and Inequalities

Equations and Inequalities Equations and Inequalities Figure 1 CHAPTER OUTLINE.1 The Rectangular Coordinate Systems and Graphs. Linear Equations in One Variable.3 Models and Applications. Comple Numbers.5 Quadratic Equations.6 Other

More information

Section 0.6: Factoring from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative

Section 0.6: Factoring from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Section 0.6: Factoring from Precalculus Prerequisites a.k.a. Chapter 0 by Carl Stitz, PhD, and Jeff Zeager, PhD, is available under a Creative Commons Attribution-NonCommercial-ShareAlike.0 license. 201,

More information

GRE Quantitative Reasoning Practice Questions

GRE Quantitative Reasoning Practice Questions GRE Quantitative Reasoning Practice Questions y O x 7. The figure above shows the graph of the function f in the xy-plane. What is the value of f (f( ))? A B C 0 D E Explanation Note that to find f (f(

More information

Common Core Algebra Regents Review

Common Core Algebra Regents Review Common Core Algebra Regents Review Real numbers, properties, and operations: 1) The set of natural numbers is the set of counting numbers. 1,2,3,... { } symbol 2) The set of whole numbers is the set of

More information

Suppose we have the set of all real numbers, R, and two operations, +, and *. Then the following are assumed to be true.

Suppose we have the set of all real numbers, R, and two operations, +, and *. Then the following are assumed to be true. Algebra Review In this appendix, a review of algebra skills will be provided. Students sometimes think that there are tricks needed to do algebra. Rather, algebra is a set of rules about what one may and

More information

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers

Chapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,

More information

Honors Math 2 Unit 5 Exponential Functions. *Quiz* Common Logs Solving for Exponents Review and Practice

Honors Math 2 Unit 5 Exponential Functions. *Quiz* Common Logs Solving for Exponents Review and Practice Honors Math 2 Unit 5 Exponential Functions Notes and Activities Name: Date: Pd: Unit Objectives: Objectives: N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of

More information

Math 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations

Math 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations

More information

Lesson 28: A Focus on Square Roots

Lesson 28: A Focus on Square Roots now Lesson 28: A Focus on Square Roots Student Outcomes Students solve simple radical equations and understand the possibility of extraneous solutions. They understand that care must be taken with the

More information

Introducing Proof 1. hsn.uk.net. Contents

Introducing Proof 1. hsn.uk.net. Contents Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction

More information

Solving Equations. A: Solving One-Variable Equations. One Step x + 6 = 9-3y = 15. Two Step 2a 3 6. Algebra 2 Chapter 1 Notes 1.4 Solving Equations

Solving Equations. A: Solving One-Variable Equations. One Step x + 6 = 9-3y = 15. Two Step 2a 3 6. Algebra 2 Chapter 1 Notes 1.4 Solving Equations Algebra 2 Chapter 1 Notes 1.4 Solving Equations 1.4 Solving Equations Topics: Solving Equations Translating Words into Algebra Solving Word Problems A: Solving One-Variable Equations The equations below

More information

Solving Quadratic & Higher Degree Equations

Solving Quadratic & Higher Degree Equations Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,

More information

MATH STUDENT BOOK. 8th Grade Unit 5

MATH STUDENT BOOK. 8th Grade Unit 5 MATH STUDENT BOOK 8th Grade Unit 5 Unit 5 More with Functions Math 805 More with Functions Introduction 3 1. Solving Equations 5 Rewriting Equations 5 Combine Like Terms 10 Solving Equations by Combining

More information

There are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality

There are two main properties that we use when solving linear equations. Property #1: Additive Property of Equality Chapter 1.1: Solving Linear and Literal Equations Linear Equations Linear equations are equations of the form ax + b = c, where a, b and c are constants, and a zero. A hint that an equation is linear is

More information

Equations and Inequalities

Equations and Inequalities Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types

More information

5.7 Translating English Sentences into Mathematical Equations and Solving

5.7 Translating English Sentences into Mathematical Equations and Solving 5.7 Translating English Sentences into Mathematical Equations and Solving Mathematical equations can be used to describe many situations in the real world. To do this, we must learn how to translate given

More information

Math 016 Lessons Wimayra LUY

Math 016 Lessons Wimayra LUY Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,

More information

Take the Anxiety Out of Word Problems

Take the Anxiety Out of Word Problems Take the Anxiety Out of Word Problems I find that students fear any problem that has words in it. This does not have to be the case. In this chapter, we will practice a strategy for approaching word problems

More information

Practice Set 1.1 Algebraic Expressions and Real Numbers. Translate each English phrase into an algebraic expression. Let x represent the number.

Practice Set 1.1 Algebraic Expressions and Real Numbers. Translate each English phrase into an algebraic expression. Let x represent the number. Practice Set 1.1 Algebraic Expressions and Real Numbers Translate each English phrase into an algebraic expression. Let x represent the number. 1. A number decreased by seven. 1.. Eighteen more than a

More information

C if U can. Algebra. Name

C if U can. Algebra. Name C if U can Algebra Name.. How will this booklet help you to move from a D to a C grade? The topic of algebra is split into six units substitution, expressions, factorising, equations, trial and improvement

More information

Sect Properties of Real Numbers and Simplifying Expressions

Sect Properties of Real Numbers and Simplifying Expressions Sect 1.7 - Properties of Real Numbers and Simplifying Expressions Concept #1 Commutative Properties of Real Numbers Ex. 1a 9.34 + 2.5 Ex. 1b 2.5 + ( 9.34) Ex. 1c 6.3(4.2) Ex. 1d 4.2( 6.3) a) 9.34 + 2.5

More information

Multiplication and Division

Multiplication and Division UNIT 3 Multiplication and Division Skaters work as a pair to put on quite a show. Multiplication and division work as a pair to solve many types of problems. 82 UNIT 3 MULTIPLICATION AND DIVISION Isaac

More information

Essential Question How can you solve an absolute value inequality? Work with a partner. Consider the absolute value inequality x

Essential Question How can you solve an absolute value inequality? Work with a partner. Consider the absolute value inequality x Learning Standards HSA-CED.A.1 HSA-REI.B.3.6 Essential Question How can you solve an absolute value inequality? COMMON CORE Solving an Absolute Value Inequality Algebraically MAKING SENSE OF PROBLEMS To

More information

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0).

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:

Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the

More information

GK- Math Review Overview

GK- Math Review Overview GK- Mathematics Resources for Some Math Questions: Kaplan et al (2015). Cliff Notes FTCE General Knowledge Test, 3 rd Edition Mander, E. (2015). FTE General Knowledge Test with Online Practice, 3 rd Edition

More information

Squaring and Unsquaring

Squaring and Unsquaring PROBLEM STRINGS LESSON 8.1 Squaring and Unsquaring At a Glance (6 6)* ( 6 6)* (1 1)* ( 1 1)* = 64 17 = 64 + 15 = 64 ( + 3) = 49 ( 7) = 5 ( + ) + 1= 8 *optional problems Objectives The goal of this string

More information

Writing and Graphing Inequalities

Writing and Graphing Inequalities 4.1 Writing and Graphing Inequalities solutions of an inequality? How can you use a number line to represent 1 ACTIVITY: Understanding Inequality Statements Work with a partner. Read the statement. Circle

More information

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS

UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS UNIT 3: MODELING AND ANALYZING QUADRATIC FUNCTIONS This unit investigates quadratic functions. Students study the structure of quadratic expressions and write quadratic expressions in equivalent forms.

More information

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x

To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression

More information