SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION

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1 Intermediate Algebra TEST 1 Spring 014 NAME: Score /100 Please print SHOW ALL YOUR WORK IN A NEAT AND ORGANIZED FASHION Course Average No Decimals No mixed numbers No complex fractions No boxed or circled answers Questions 1 30 are 1 pt. each Material presented in red is the minimal answer to the question. Material in blue is either altenate answers or explanations of basics. 1. T F 5x 3x + 7 is an equation.. T F A number (or numbers) that makes an equation true when substituted for the variable (or variables) is called a solution of the equation. 3. T F If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation. 4. T F A number is written in scientific notation if it is written as the product of a real number and an integer power of T F The product of two numbers written in scientific notation is the number obtained by multiplying the coefficients according to normal arithmetic properties and multiplying the powers of 10 according to the laws of exponents. 6. T F A term is a letter, a number, or a product of letters and numbers. 7. T F The variable part of a term is called the coefficient of the term. 8. T F If both sides of an equation are multiplied by the same expression, the resulting equation is equivalent to the original equation. 9. T F If m, n and a are real numbers then a m a n = a m + n. 10. T F If a is not zero then a 0 = T F If both sides of an inequality are multiplied by a non-zero real number, the resulting inequality is equivalent to the original inequality. 1. T F If a, m and n are real numbers then (a m ) n = a m+n. 13. T F The process to solve a linear equation in one variable is to generate a sequence of equations each equivalent to the previous equation until a simplest equation is obtained. 14. T F The product of two terms is the term obtained by multiplying the coefficients according to normal arithmetic properties and multiplying the variables according to the laws of exponents. 15. Two real numbers a and b are equal if they represent the same point on the number line. 16. The real number a is greater than b if a is to the right of b on the number line. 17. A binary relation is a comparison of two operands. 18. A binary operation is a calculation involving two operands. 19. In the exponential expression a n, n is the exponent 0. The opposite of a real number is the number which is equidistant from zero but on the opposite side of zero. 1. The set of Integers consists of the Whole Numbers with their opposites adjoined.. The set of Rational numbers consists of all numbers which can be written as fractions. 3. Write the formula for the area of a triangle A 1 bh.(remember a formula must be an equation) 1

2 4. Write the formula for the circumference of a circle. C D Mathematics is case sensitive. Moreover unless there is a conflict, area is designated with a capital A, volume with a capital V, and linear quantities are usually designated by lower case letters. Notable exceptions to this rule of thumb is diameter and circumference (linear measures). Diameter is designated with a captial D and circumference is designated with a capital C. The standard formulas related to elementary geometric shapes and solids should be memorized and repeated in their standard form. 5. A conditional equation is an equation which is true when some real numbers are substituted for the variables and is false when some real numbers are substituted for the variables. 6. Two equations are equivalent if they have the same solution sets. EQUATIONS CANNOT BE EQUAL. EQUATIONS CAN BE EQUIVALENT. SETS CANNOT BE EQUIVALENT. SETS CAN BE EQUAL. 7. A polynomial is a term or a sum of terms in which all variables have whole number exponents. 8. Two terms are called like terms if they have the same variables with the same exponents. 9. In the expression 5 4, the base is 5. If a negative number is the base of an exponential expression it must be enclosed in parenthsis. Textbook Page The opposite of 4 is In the following multiple choice questions, any number of choices may be correct. In each question at least one choice is correct. Circle ALL correct choices. 31. (3 pts) Which of the following are solutions of x < 1? a. 1 b. 0 c. 1 d. 5 e. 3 f. 3. (4 pts) Which of the following are linear equations in one variable? a. 3x + b. y = 9x 5 c. 3x + 1 = x 7 d. 8x 5 = 0 e. x + 5x + 6 = 0 f. x 5x 3 7 g. 5x 1 0 h x 33. (3 pts) Insert the correct symbol in the to make the statement true. You should choose from these symbols: = < > a. 9 {x x is an integer}

3 b. {3} {x x is an integer} c. 4 {x x is an irrational number} d. 3 {x x is a rational number} e. (3, 5) = {x 3 < x <5} f. {3,5} (3, 5) 34. (4 pts) Complete the following algebraic definition of absolute value. x if x 0 x x if x (1 pts) Complete the following statement of the Fundamental Property of Fractions If a represents a real number and b and c represent non-zero real numbers then ac a bc b. Textbook Page 17. Generalized version Page (1 pts) Complete the following statement of the Distributive Property. If a, b, and c are real numbers, then a(b + c) = ab + ac. 37. (3 pts) Complete the statement of The Law of Trichotomy If a and b are real numbers then exactly one of the following is true: i. a < b ii. a = b iii. a > b Problems are each worth 4 points. 38. Write the set {x 9 < x < 1} in interval notation. (9, 1) 39. Sketch the graph of the interval (, 5). Intervals are segments of the real number line. Therefore their graphs must be on the real number line. When the statement reads interval (, 5). then it is clear that (, 5) does not represent a point in the Cartesian plane. When drawing the graph of an interval on the real number line it is undesirable to include a whole bunch of natural numbers in the graph. See the above example. Read the material in Section RP5 on the website. 40. Write the interval ( 3, ] in set builder notation. x 3 x 41. Compute the product ( )( ) ( 4)( ) 8 About the only place we use X to indicate multiplication in algebra is when discussing scientific notation. Stop using it elsewhere. In algebra we indicate multiplication by writing the factors next to each other (this is called juxtaposition). If there is any chance of confusion, the individual factors are enclosed in grouping symbols. Common denominators are not required to multiply fractions. When you bring common denominators into a discussion of multiplication (or division), I must conclude you lack even the most fundamental understanding of the arithmetic of fractions. 3

4 4. Compute the quotient or ( 3)( ) ( 4)( 5) Compute the sum Don t write obvious falsehoods like true. That destroys the logic chain leading to the desired conclusion that the sum is 9/. Don t write errors like not 9 and 4 4 means nothing. 44. Compute the sum or The two green = signs are simply not Do not write What your really mean and what is mathematically correct is 9 3 ( 9)( 3) or 9 4 ( 9)( 4) Observe the use of the = symbol in the above computations, read the book, read the website. If two expressions are equal you should indicate that fact with the use of the equal symbol. This symbol ( ) has no place in any algebra work that I know of. It is used in logic and in functional analysis but we are years from that Write in scientific notation Write X 10 4 in standard notation , Compute the product (7 X 10 4 )(3 X 10 9 ). Write the product in scientific notation ( 7)( 3) ( 10 )( 10 ) Compute the sum (87 X 10 4 ) + ( 3 X 10 4 ). Write the sum in standard notation ( 3) , What is the solution set for the equation 3x + 4 = 7x 1? 3x + 4 = 7x 1 4x + 4 = 1 4x = 16 4

5 x = 4 The solutions set is {4} When you stop with the statement x = 4 (even though the instructions asked for the solution set), I am forced to believe that: you do not understand the difference between solution and solution set or you do not realize that you have found the solution set or both. It may have been acceptable (I doubt it) in previous courses to accept x = 4 as the answer, but that is not the case going forward. We expect you to GROW in your knowledge and understanding of algebra. When solving equations or inequalities do not write something like the following: 3x + 4 = 7x 1-3x - 3x 4x + 4 = x x = 4 The solutions set is {4} The material shown in blue should not be included. It may have been a good idea when first learning about solving equations (I doubt it) but we must strive to grow. 50. Consider the equation x 5 x 4 + x = 5. a. Is 1 a solution of the above equation? NO b. Justify your answer to Part a. (This justification MUST contain some words) 1 is not a solution because if x is replaced with 1 a FALSE statement (=5) results. A solution is a number which produces a TRUE statement when substituted for the variable. The only way to determine if a number is a solution of an equation or an inequality is to determine if that number satisfies the definition of solution. That definition is: A NUMBER (OR NUMBERS) THAT MAKES AN EQUATION TRUE WHEN SUBSTITUTED FOR THE VARIABLE (OR VARIABLES) IS A SOLUTION OF THE EQUATION. Statements like the following are meaningless: I substituted 1 in the equation and the solutions were not equal. Look at the definition of solution and it becomes clear that substituting 1 into the equation does not produce solutions. I substituted 1 into the equation and both sides were not equal. This is meaningless even in a grammatical sense. Think about what you write! AS IS ALWAYS THE CASE: COMMUNICATION SKILLS WERE LACKING To learn how to write the mathematics, you should study and try to imitate the writing style of an accomplished math teacher -- Elayn Martin-Gay To see how the process of solving equations should be written, take a look at the examples on Pages 85 and 853. You should know that what appears in blue is for instructional purposes and what is in black is the presentation you should strive for. 5

6 The stuff in blue should not appear in your solutions because you are not trying to "teach". Examples 6 and 7 on Page 80 show how to use the = symbol in work like simplifying. Note how your author depends on The Transitive Property of Equality. Examples 1-8 on pages are good illustrations of writing. But you must keep in mind the blue text is for instructional purposes only. Look at Example 1 on Page 315 to see how your author uses the = symbol in her presentation of simplification problems involving exponents. Look at Example 1 on Page 441 to see how your author answers related to the domain of a rational expression. Note the use of words. Note she concludes the domain is a set and she chooses to use set builder notation to describe that set. On the test and in your homework you MUST write like those examples. Observe that my comment attached to Question 39 is in stark contrast with your author s examples of graphs on the number line. We do not disagree on what constitutes such a graph, however I expect you can GROW beyond the elementary presentation that includes a bunch of integers. Part of my reasoning is that such graphs foster the idea that the only numbers on the number line (or in the interval) are integers. 6

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,