Appendix A Review of Fundamental Concepts of Algebra

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1 Appendi A Review of Fundamental Concepts of Algebra Section A.: Real Numbers and Their Properties Objective : Represent and classify real numbers. Identify which of the following numbers are natural numbers ,, 7, 0, 5, 0., 7, 0 5 0, 5 5, 7 5, 5 7, 7 5, 5, 5. Write the rational number as the ratio of two integers Identify the numbers that are not integers. 4,,., 0, 4., 4, Use a calculator to find the decimal form of the rational number. If it is a nonterminating decimal, write the repeating pattern. 4 Objective : Order real numbers and use inequalities 5. Use inequality notation to describe the set. Julia scored at least points. k k k k 6. Identify the verbal description of the interval., 0 g All real numbers greater than 0 and less than or equal to All real numbers greater than 0 and less than All real numbers greater than or equal to 0 and less than or equal to All real numbers greater than or equal to 0 and less than Copyright Houghton Mifflin Company. All rights reserved. 45

2 Testbank Appendi A Review of Fundamental Concepts of Algebra 7. Place the correct symbol (,, or =) between the pair of real numbers Verbally describe the subset of real numbers represented by the inequality. Then sketch the subset on the real number line. State whether the interval is bounded or unbounded. 4 Objective : Find the absolute values of real numbers and find the distance between two real numbers 9. Caleb is on a mountain,46 feet above sea level. Leslie is in a submarine 79 feet below sea level. Which of the following can be used to find the difference between Caleb s elevation and Leslie s elevation?, ,46, ,46 0. Identify the epression that corresponds to the distance shown by the bar Order the values from smallest to largest. 5,, 7,. Evaluate the epression. M M Objective 4: Evaluate algebraic epressions Evaluate the epression for the given value of.. 5 F H G I 7 K J b g for Copyright Houghton Mifflin Company. All rights reserved.

3 Testbank Section A.: Real Numbers and Their Properties Evaluate the epression for the given value of for None of these 5. for = for 5 Objective 5: Use the basic rules and properties of algebra 7. Identify which of the following is an eample of the Commutative Property of Multiplication. b g b g Identify the rule of algebra illustrated by the equation Commutative Property of Addition Associative Property of Addition Distributive Property Additive Identity Property b g 9. Give an eample of the Multiplicative Inverse Property. 0. Perform the operation. (Write the fractional answer in simplest form.) 9 Section A.: Eponents and Radicals Objective : Use properties of eponents. Simplify the epression. F HG I KJ Evaluate the epression Copyright Houghton Mifflin Company. All rights reserved. 47

4 Testbank Appendi A Review of Fundamental Concepts of Algebra. Evaluate each epression. (a) (b) 6 9 b g (c) 7 (d) Simplify the epression. e e 4 5 y 5 y j j Objective : Use scientific notation to represent real numbers 5. Identify the number written in decimal form ,400,000 4,000,000,40, Find the product written in scientific notation e je j Write the number in scientific notation. 7. A virus takes up a volume of approimately cubic centimeter. 8. In 995, Cambodia had a population of about 0,70,000 people. Objective : Use properties of radicals 9. Perform the operation and simplify the epression. F I 5 H K Evaluate the epression without using a calculator Copyright Houghton Mifflin Company. All rights reserved.

5 Testbank Section A.: Eponents and Radicals Evaluate the epression without using a calculator.. 9. Use a calculator to approimate the number. (Round your answer to three decimal places.) 6 7 Objective 4: Simplify and combine radicals. Simplify by removing all possible factors from each radical Simplify the epression Objective 5: Rationalize denominators and numerators 7. Rationalize the denominator of the epression. Then simplify the answer Rationalize the numerator of the epression. Then simplify the answer Copyright Houghton Mifflin Company. All rights reserved. 49

6 Testbank Appendi A Review of Fundamental Concepts of Algebra Rationalize the numerator of the epression. Then simplify the answer Rationalize the denominator of the epression. Then simplify the answer. Objective 6: Use properties of rational eponents 4. Perform the operation and simplify Write the epression as a single radical. Then simplify your answer Use a calculator to approimate the number. (Round your answer to three decimal places.) b g Evaluate the epression without using a calculator. 4 b g Section A.: Polynomials and Factoring Objective : Write polynomials in standard form 45. Identify the polynomial written in standard form Not a polynomial 50 Copyright Houghton Mifflin Company. All rights reserved.

7 Testbank Section A.: Polynomials and Factoring 46. Identify the polynomial written in standard form Write a polynomial that fits the description. (There are many correct answers.) A second-degree polynomial with leading coefficient Write the polynomial in standard form. 5 Objective : Add, subtract, and multiply polynomials Perform the operations and identify the result written in standard form. e a a 6a 5a j a a 5a 4a a 4a 0a 6a a 0a a 8a a 4a 4a b gb g e j b g e j 5. 5 k k 6 9 k 5 k b g b gb g Objective : Use special products to multiply polynomials Multiply or find the special product. b g y 6 4y 6y 6 48y6y 6 6y 6 4y6y Copyright Houghton Mifflin Company. All rights reserved. 5

8 Testbank Appendi A Review of Fundamental Concepts of Algebra Multiply or find the special product. e 54. y j y y y 8 y 6y y y y y y F I HG K JF HG y 4 y 5 5 b g b g I K J Objective 4: Remove common factors from polynomials Factor out the common factor F I HG K J e 4j e e j j e j 8e 5 5j 8 8 e 4 5 j 7 e j b g b g Copyright Houghton Mifflin Company. All rights reserved.

9 Testbank Section A.: Polynomials and Factoring Objective 5: Factor special polynomial forms Completely factor the epression. 6. 5n 60n6 b gb g b5n 6g b5n6gb5ng b5n 6g 5n6 5n y b g b9 4ygb94yg b ygb yg b9 4ygb94yg 9 4y Objective 6: Factor trinomials as the product of two binomials Factor the trinomial. b g b g b g bf ggby7gby g bf ggby7gbyg by7gbyg bf gge y y j f g y 6y f g 7 f g b g b g b gb g bgb9g bgb9g bgb9g v v 40v Copyright Houghton Mifflin Company. All rights reserved. 5

10 Testbank Appendi A Review of Fundamental Concepts of Algebra Objective 7: Factor polynomials by grouping Factor by grouping b gb g e jb 7g e jb 7g bgb 7g e je jb g e je j e je j e je jbg Section A.4: Rational Epressions Objective : Find domains of algebraic epressions Find the domain of the epression All real numbers such that 6 All real numbers such that 6 All real numbers such that 0 All real numbers such that All real numbers such that 0 All real numbers All real numbers such that 0 All real numbers such that 54 Copyright Houghton Mifflin Company. All rights reserved.

11 Testbank Section A.4: Rational Epressions Find the domain of the epression Objective : Simplify rational epressions Write the rational epression in simplest form. 77., 0,, 0,, 0,, 0, b g, 6 b b g, 5 b g, g, Copyright Houghton Mifflin Company. All rights reserved. 55

12 Testbank Appendi A Review of Fundamental Concepts of Algebra Objective : Add, subtract, multiply, and divide rational epressions Perform the operation(s) and simplify y 7y0 y y y 7y0 5 y b g y y b5 ygby5g bygbyg 5 y y5 b gb g b g y y b5 ygby5g y b5 ygby5g Objective 4: Simplify comple fractions Simplify the comple fraction b gb g 4 b gb g 4 4 b gb g b gb g 7 b gb g b gb g e j 5 e6 j 6 / / 6 4 e6 j / 4 e6 j / 6 None of these e6 j / 56 Copyright Houghton Mifflin Company. All rights reserved.

13 Testbank Section A.4: Rational Epressions Simplify the comple fraction Section A.5: Solving Equations Objective : Identify different types of equations 89. Which of the following is an identity? b g 7b g 7 b g 7b g Which of the following is a conditional equation? F HG I K J Determine whether the equation is an identity or a conditional equation. b5g b6g 9. Determine whether the equation is a conditional or an identity. 0 5 b g Objective : Solve linear equations in one variable Solve the equation and check your solution. b g Copyright Houghton Mifflin Company. All rights reserved. 57

14 Testbank Appendi A Review of Fundamental Concepts of Algebra Solve the equation and check your solution. b g b g Find the solution to the equation b g 96. Solve the equation and check your solution. j 7 9 Objective : Solve quadratic equations by factoring, etracting square roots, completing the square, and using the Quadratic Formula 97. Use the quadratic formula to solve the equation Solve the equation by factoring , 8,, 8, Solve the equation by etracting square roots. b g 00. Use the quadratic formula to solve the equation. 80 Objective 4: Solve polynomial equations of degree three or greater Find all real solutions of the equation. Check your solutions in the original equation ,,,, Copyright Houghton Mifflin Company. All rights reserved.

15 Testbank Section A.5: Solving Equations Find all real solutions of the equation. Check your solutions in the original equation b gb gb gb g b gb gb gb g b gb gb gb g b gb gb gb g Objective 5: Solve equations involving radicals Find all real solutions of the equation. Check your solutions in the original equation = No solution 54 b g 75 or or b g b g = Objective 6: Solve equations involving absolute values Find all real solutions of the equation. Check your solutions in the original equation , 4, 9, ,, 4 No solution. 4 Copyright Houghton Mifflin Company. All rights reserved. 59

16 Testbank Appendi A Review of Fundamental Concepts of Algebra Find all real solutions of the equation. Check your solutions in the original equation Section A.6: Solving Inequalities Objective : Represent solutions of linear inequalities in one variable. Which is the graph of the inequality? < < Which of the following intervals represents 6 < 6? Graph. b g, 6, , 6g 6, 6 b Copyright Houghton Mifflin Company. All rights reserved.

17 Testbank Section A.6: Solving Inequalities Objective : Solve linear inequalities in one variable Solve the inequality None of these b g Objective : Solve inequalities involving absolute values. Solve the inequality. 5 >, < < < >, < < < 5 5. Which is the graph of the solution for the inequality? Copyright Houghton Mifflin Company. All rights reserved. 6

18 Testbank Appendi A Review of Fundamental Concepts of Algebra. Graph the solution for the inequality Solve: Objective 4: Solve polynomial and rational inequalities 5. Solve the inequality and give the solution in interval notation F HG F HG I K J 4 6, 4+ 6, , O Q P, , j 4, Solve the inequality and graph the solution on the real number line., 4 F HG L NM I K J e 6 6 O Q P j , Solve the inequality Copyright Houghton Mifflin Company. All rights reserved.

19 Testbank Section A.6: Solving Inequalities 8. Solve the inequality and give the solution in interval notation. 9 6 b gb g 0 4 Section A.7: Errors and the Algebra of Calculus Objective : Avoid common algebraic errors 9. Identify the missing factor which, if placed in the parentheses, would make the equation true. b 5g 65 5b 5g 5 b 5g b g 5 65 b g b g 6 0. Which equation is correct? Correct the error. 4 6 b g 4b 4g If there is an error in the equality, correct the error. If no error eists, write True Objective : Recognize and use algebraic techniques that are common in calculus Insert the required factor in the parentheses.. e 4 7 j = b g b9 g e j 7 7 None of these Copyright Houghton Mifflin Company. All rights reserved. 6

20 Testbank Appendi A Review of Fundamental Concepts of Algebra Insert the required factor in the parentheses = 49 b g b g 49 5, 8 6, , 6 8, Simplify the epression. b7g b5g b5g b7g 4 4 b L N M 7 g O Q P 4 6. Write the fraction as a sum of two or more terms Section A.8: Graphical Representation of Data Objective : Plot points in the Cartesian plane b g b g b g 7. Identify the graph of the points A,, B,, and C,. B y C B C y A A y y A A C B C B 64 Copyright Houghton Mifflin Company. All rights reserved.

21 8. Determine the quadrant in which, y 0 and y0 b Testbank Section A.8: Graphical Representation of Data gis located so that the conditions are satisfied. Quadrant I Quadrant II Quadrant III Quadrant IV 9. Name the coordinates of point A and the quadrant in which A is located. y A Find the coordinates of the point that is located units to the right of the y-ais and units above the -ais. Objective : Use the Distance Formula to find the distance between two points 4. Find the length of the hypotenuse of the triangle. y Find the distance between the points. F I HG K J and e 0j 6 Copyright Houghton Mifflin Company. All rights reserved. 65

22 Testbank Appendi A Review of Fundamental Concepts of Algebra Find the distance between the points. b g b g 4. 6, 4 and 7, Verify that the triangle with vertices S 7, 0, triangle. b g T b, 0g, and U b, 4 4g is an isosceles Objective : Use the Midpoint Formula to find the midpoint of a line segment 45. Find the midpoint of the line segment connecting b5, 5gand b 0, 8 g. b g F 5 HG 5, I K J, F 5 I, HG K J b5, g 46. Find the midpoint of AB. y 0 B A 0 0, 47. M, 0 b g b, g b, g b, b g is the midpoint of RS. If S has coordinates b5, 5g, find the coordinates of R. g 66 Copyright Houghton Mifflin Company. All rights reserved.

23 Testbank Section A.8: Graphical Representation of Data 48. Find the midpoint of the line segment connecting the two points. Then show that the midpoint is the same distance from each point. y 0 E 0 0 F 0 Objective 4: Use a coordinate plane to model and solve real-life problems 49. A highway map of Ohio has a coordinate grid superimposed on top of the state. Springfield is at point b, 4g and Cleveland is at point b5, g. The Springfield History Club is going to Cleveland to see the Rock and Roll Hall of Fame. The map shows a highway rest area halfway between the cities. What are the coordinates of the rest area? What is the distance between Springfield and Cleveland? (one unit = 7. miles) 7 Rest area =, Cleveland = 7 miles 7 Rest area =, Cleveland = 08 miles F HG F HG I K J I K J F HG 7 Rest area =, Cleveland = 89 miles F HG 7 Rest area =, Cleveland = 9 miles I K J I K J Copyright Houghton Mifflin Company. All rights reserved. 67

24 Testbank Appendi A Review of Fundamental Concepts of Algebra 50. On a street map, the side of each square represents mile. The local Q-Mart is located at,, and the Circle Cineple is located at b6, 0g. A diagonal street runs directly between the two locations. Approimately how far is it from Q-Mart to Circle Cineple? b g Circle Cineple Q-Mart 9 miles 7 miles 89 miles miles 5. Juanita made a sketch of a circular pool on a graph grid. On the graph the diameter of the pool has endpoints at b6, g and b 8, 5g. What are the coordinates of the center of the pool? 5. You live in Carson City, Nevada, which has approimate latitude-longitude coordinates of b9 N, 0 Wg and your friend lives in Ottawa, Ohio, with coordinates of b4 N, 84 Wg. You plan to meet halfway between the two cities. Find the coordinates of the halfway point. 68 Copyright Houghton Mifflin Company. All rights reserved.

Precalculus Notes: Unit P Prerequisite Skills

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