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Unformatted text preview: CK-12 FOUNDATION Algebra I Gloag Gloag CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, webbased collaborative model termed the “FlexBook,” CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning. Copyright © 2010 CK-12 Foundation, Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC-by-NC-SA) License ( . org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Specific details can be found at . Printed: August 3, 2010 Authors Andrew Gloag, Anne Gloag i Contents 1 Equations and Functions 1 1.1 Variable Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Patterns and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 Equations and Inequalities 31 1.5 Functions as Rules and Tables . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.6 Functions as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1.7 Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 1.8 Problem-Solving Strategies: Make a Table and Look for a Pattern . . . . . 80 1.9 Additional Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Real Numbers 95 2.1 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.2 Addition of Rational Numbers 2.3 Subtraction of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . 114 2.4 Multiplication of Rational Numbers 2.5 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.6 Division of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 142 2.7 Square Roots and Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . 148 2.8 Problem-Solving Strategies: Guess and Check, Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . 106 3 Equations of Lines . . . . . . . . . . . . . . . . . . . . . . 122 . . . . . . . 160 171 iii 3.1 One-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 3.2 Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 3.3 Multi-Step Equations 3.4 Equations with Variables on Both Sides . . . . . . . . . . . . . . . . . . . . 203 3.5 Ratios and Proportions 3.6 Scale and Indirect Measurement 3.7 Percent Problems 3.8 Problem Solving Strategies: Use a Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 . . . . . . . . . . . . . . . . . . . . . . . . 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 . . . . . . . . . . . . . . . . . . 250 4 Graphs of Equations and Functions 259 4.1 The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 4.2 Graphs of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4.3 Graphing Using Intercepts 4.4 Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 4.5 Graphs Using Slope-Intercept Form 4.6 Direct Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 4.7 Linear Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 4.8 Problem-Solving Strategies - Graphs . . . . . . . . . . . . . . . . . . . . . . 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 . . . . . . . . . . . . . . . . . . . . . . 314 5 Writing Linear Equations 359 5.1 Linear Equations in Slope-Intercept Form . . . . . . . . . . . . . . . . . . . 359 5.2 Linear Equations in Point-Slope Form . . . . . . . . . . . . . . . . . . . . . 375 5.3 Linear Equations in Standard Form 5.4 Equations of Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . 397 5.5 Fitting a Line to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 5.6 Predicting with Linear Models 5.7 Problem Solving Strategies: Use a Linear Model . . . . . . . . . . . . . . . . . . . . . . 385 . . . . . . . . . . . . . . . . . . . . . . . . . 430 . . . . . . . . . . . . . . . 442 6 Graphing Linear Inequalities; Introduction to Probability 6.1 453 Inequalities Using Addition and Subtraction . . . . . . . . . . . . . . . . . . 453 iv 6.2 Inequalities Using Multiplication and Division . . . . . . . . . . . . . . . . . 460 6.3 Multi-Step Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 6.4 Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 6.5 Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 6.6 Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 6.7 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . 507 7 Solving Systems of Equations and Inequalities 525 7.1 Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . 525 7.2 Solving Linear Systems by Substitution 7.3 Solving Linear Systems by Elimination through Addition or Subtraction . . 547 7.4 Solving Systems of Equations by Multiplication . . . . . . . . . . . . . . . . 554 7.5 Special Types of Linear Systems 7.6 Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . 537 . . . . . . . . . . . . . . . . . . . . . . . . 565 . . . . . . . . . . . . . . . . . . . . . . . . . 578 8 Exponential Functions 595 8.1 Exponent Properties Involving Products . . . . . . . . . . . . . . . . . . . . 595 8.2 Exponent Properties Involving Quotients 8.3 Zero, Negative, and Fractional Exponents . . . . . . . . . . . . . . . . . . . 613 8.4 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622 8.5 Exponential Growth Functions 8.6 Exponential Decay Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 644 8.7 Geometric Sequences and Exponential Functions 8.8 Problem-Solving Strategies . . . . . . . . . . . . . . . . . . . 604 . . . . . . . . . . . . . . . . . . . . . . . . . 633 . . . . . . . . . . . . . . . 654 . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 9 Factoring Polynomials; More on Probability 671 9.1 Addition and Subtraction of Polynomials . . . . . . . . . . . . . . . . . . . 671 9.2 Multiplication of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 683 9.3 Special Products of Polynomials 9.4 Polynomial Equations in Factored Form . . . . . . . . . . . . . . . . . . . . 703 . . . . . . . . . . . . . . . . . . . . . . . . 696 v 9.5 Factoring Quadratic Expressions . . . . . . . . . . . . . . . . . . . . . . . . 715 9.6 Factoring Special Products 9.7 Factoring Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 . . . . . . . . . . . . . . . . . . . . . . . 736 10 Quadratic Equations and Quadratic Functions 749 10.1 Graphs of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . 749 10.2 Quadratic Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . 770 10.3 Quadratic Equations by Square Roots . . . . . . . . . . . . . . . . . . . . . 791 10.4 Solving Quadratic Equations by Completing the Square . . . . . . . . . . . 801 10.5 Solving Quadratic Equations by the Quadratic Formula . . . . . . . . . . . 816 10.6 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831 10.7 Linear, Exponential and Quadratic Models . . . . . . . . . . . . . . . . . . 839 10.8 Problem Solving Strategies: Choose a Function Model . . . . . . . . . . . . 861 11 Algebra and Geometry Connections; Working with Data 877 11.1 Graphs of Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . . 877 11.2 Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 11.3 Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 11.4 The Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . 928 11.5 Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . . . 941 11.6 Measures of Central Tendency and Dispersion . . . . . . . . . . . . . . . . . 954 11.7 Stem-and-Leaf Plots and Histograms . . . . . . . . . . . . . . . . . . . . . . 968 11.8 Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 12 Rational Equations and Functions; Topics in Statistics 12.1 Inverse Variation Models 995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995 12.2 Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 12.3 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1028 12.4 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1038 12.5 Multiplication and Division of Rational Expressions vi . . . . . . . . . . . . . 1047 12.6 Addition and Subtraction of Rational Expressions 12.7 Solutions of Rational Equations 12.8 Surveys and Samples . . . . . . . . . . . . . . 1055 . . . . . . . . . . . . . . . . . . . . . . . . 1071 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085 vii viii Chapter 1 Equations and Functions 1.1 Variable Expressions Learning Objectives • Evaluate algebraic expressions. • Evaluate algebraic expressions with exponents. Introduction – The Language of Algebra Do you like to do the same problem over and over again? No? Well, you are not alone. Algebra was invented by mathematicians so that they could solve a problem once and then use that solution to solve a group of similar problems. The big idea of algebra is that once you have solved one problem you can generalize that solution to solve other similar problems. In this course, we’ll assume that you can already do the basic operations of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷ ) occur. In algebra, numbers (and sometimes processes) are denoted by symbols (such as x, y, a, b, c, . . .). These symbols are called variables. The letter x, for example, will often be used to represent some number. The value of x, however, is not fixed from problem to problem. The letter x will be used to represent a number which may be unknown (and for which we may have to solve) or it may even represent a quantity which varies within that problem. Using variables offers advantages over solving each problem “from scratch”: • It allows the general formulation of arithmetical laws such as a + b = b + a for all real 1 numbers a and b. • It allows the reference to “unknown” numbers, for instance: Find a number x such that 3x + 1 = 10. • It allows short-hand writing about functional relationships such as, “If you sell x tickets, then your profit will be 3x − 10 dollars, or f (x) = 3x − 10,” where “f ” is the profit function, and x is the input (i.e. how many tickets you sell). Example 1 Write an algebraic expression for the perimeter and area of the rectangle as follows. To find the perimeter, we add the lengths of all 4 sides. We can start at the top-left and work clockwise. The perimeter, P , is therefore: P =l+w+l+w We are adding 2 l’s and 2 w’s. Would say that: P =2× l+2× w You are probably familiar with using · instead of × for multiplication, so you may prefer to write: P =2·l+2·w It’s customary in algebra to omit multiplication symbols whenever possible. For example, 11x means the same thing as 11 · x or 11 × x. We can therefore write the expression for P as: P = 2l + 2w 2 Area is length multiplied by width. In algebraic terms we get the expression: A=l×w → A=l·w → A = lw Note: An example of a variable expression is 2l + 2w; an example of an equation is P = 2l + 2w. The main difference between equations and expressions is the presence of an equals sign (=). In the above example, there is no simpler form for these equations for the perimeter and area. They are, however, perfectly general forms for the perimeter and area of a rectangle. They work whatever the numerical values of the length and width of some particular rectangle are. We would simply substitute values for the length and width of a real rectangle into our equation for perimeter and area. This is often referred to as substituting (or plugging in) values. In this chapter we will be using the process of substitution to evaluate expressions when we have numerical values for the variables involved. Evaluate Algebraic Expressions When we are given an algebraic expression, one of the most common things we will have to do with it is evaluate it for some given value of the variable. The following example illustrates this process. Example 2 Let x = 12. Find the value of 2x − 7. To find the solution, substitute 12 for x in the given expression. Every time we see x we will replace it with 12. Note: At this stage we place the value in parentheses: 2x − 7 = 2(12) − 7 = 24 − 7 = 17 The reason we place the substituted value in parentheses is twofold: 1. It will make worked examples easier for you to follow. 2. It avoids any confusion that would arise from dropping a multiplication sign: 2 · 12 = 2(12) ̸= 212. Example 3 3 Let x = −1. Find the value of −9x + 2. Solution −9(−1) + 2 = 9 + 2 = 11 Example 4 Let y = −2. Find the value of 7 y − 11y + 2. Solution 7 1 − 11(−2) + 2 = −3 + 22 + 2 (−2) 2 1 = 24 − 3 2 1 = 20 2 Many expressions have more than one variable in them. For example, the formula for the perimeter of a rectangle in the introduction has two variables: length (l) and width (w). In these cases be careful to substitute the appropriate value in the appropriate place. Example 5 The area of a trapezoid is given by the equation A = h2 (a + b). Find the area of a trapezoid with bases a = 10 cm, b = 15 cm and height h = 8 cm. To find the solution to this problem we simply take the values given for the variables, a, b and h, and plug them in to the expression for A: 4 h (a + b) 2 8 A = (10 + 15) 2 A = 4(25) = 100 Substitute 10 for a, 15 for b and 8 for h. A= Evaluate piece by piece. (10 + 15) = 25; 8 =4 2 Solution: The area of the trapezoid is 100 square centimeters. Example 6 Find the value of 1 (5x 9 + 3y + z) when x = 7, y = −2 and z = 11. Let’s plug in values for x, y and z and then evaluate the resulting expression. 1 (5(7) + 3(−2) + (11)) 9 1 (35 + (−6) + 11) 9 1 40 (40) = ≈ 4.44 9 9 Evaluate the individual terms inside the parentheses. Combine terms inside parentheses. Solution ≈ 4.44(rounded to the nearest hundredth) Example 7 The total resistance of two electronics components wired in parallel is given by R1 R2 R1 + R2 where R1 and R2 are the individual resistances (in ohms) of the two components. Find the combined resistance of two such wired components if their individual resistances are 30 ohms and 15 ohms. Solution R1 R2 R1 + R2 450 (30)(15) = = 10 ohms 30 + 15 45 Substitute the values R1 = 30 and R2 = 15. The combined resistance is 10 ohms. 5 Evaluate Algebraic Expressions with Exponents Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for repeated multiplication. For example: 2 · 2 = 22 2 · 2 · 2 = 23 The exponent stands for how many times the number is used as a factor (multiplied). When we deal with integers, it is usually easiest to simplify the expression. We simplify: 22 = 4 and 23 = 8 However, we need exponents when we work with variables, because it is much easier to write x8 than x · x · x · x · x · x · x · x. To evaluate expressions with exponents, substitute the values you are given for each variable and simplify. It is especially important in this case to substitute using parentheses in order to make sure that the simplification is done correctly. Example 8 The area of a circle is given by the formula A = πr2 . Find the area of a circle with radius r = 17 inches. Substitute values into the equation. 6 A = πr2 A = π(17)2 Substitute 17 for r. π · 17 · 17 = 907.9202 . . . Round to 2 decimal places. The area is approximately 907.92 square inches. Example 9 Find the value of 5x2 − 4y for x = −4 and y = 5. Substitute values in the following: 5x2 − 4y = 5(−4)2 − 4(5) = 5(16) − 4(5) = 80 − 20 = 60 Substitute x = −4 and y = 5. Evaluate the exponent (−4)2 = 16. Example 10 Find the value of 2x2 − 3x2 + 5, for x = −5. Substitute the value of x in the expression: 2x2 − 3x2 + 5 = 2(−5)3 − 3(−5)2 + 5 Substitute − 5 for x. = 2(−125) − 3(25) + 5 Evaluate exponents (−5)3 = (−5)(−5)(−5) = −125 and (−5)2 = (− = −250 − 75 + 5 = −320 Example 11 Find the value of x2 y 3 x3 +y 2 , for x = 2 and y = −4. Substitute the values of x and y in the following. (2)2 (−4)3 x2 y 3 = Substitute 2 for x and − 4 for y. x3 + y 2 (2)3 + (−4)2 4(−64) −256 −32 = = Evaluate expressions : (2)2 = (2)(2) = 4 and (2)3 = (2)(2)(2) = 8. 8 + 16 24 3 (−4)2 = (−4)(−4) = 16 and (−4)3 = (−4)(−4)(−4) = −64. 7 Example 12 The height (h) of a ball in flight is given by the formula: h = −32t2 + 60t + 20, where the height is given in feet and the time (t) is given in seconds. Find the height of the ball at time t = 2 seconds. Solution h = −32t2 + 60t + 20 = −32(2)2 + 60(2) + 20 = −32(4) + 60(2) + 20 = 12 feet Substitute 2 for t. Review Questions Write the following in a more condensed form by leaving out a multiplication symbol. 1. 2 × 11x 2. 1.35 · y 3. 3 × 14 4. 14 · z Evaluate the following expressions for a = −3, b = 2, c = 5 and d = −4. 5. 2a + 3b 6. 4c + d 7. 5ac − 2b 2a 8. c−d 9. 3b d a−4b 10. 3c+2d 1 11. a+b 12. ab cd Evaluate the following expressions for x = −1, y = 2, z = −3, and w = 4. 13. 8x3 2 14. 5x 6z 3 15. 3z 2 − 5w2 16. x2 − y 2 8 17. 18. 19. 20. 21. z 3 +w3 z 3 −w3 2 2x − 3x2 + 5x − 4 4w3 + 3w2 − w + 2 3 + z12 The weekly cost C of manufacturing x remote controls is given by the formula C = 2000 + 3x, where the cost is given in dollars. (a) What is the cost of producing 1000 remote controls? (b) What is the cost of producing 2000 remote controls? 22. The volume of a box without a lid is given by the formula: V = 4x(10 − x)2 where x is a length in inches and V is the volume in cubic inches. (a) What is the volume when x = 2? (b) What is the volume when x = 3? Review Answers 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 22x 1.35y 3 4 z 4 0 16 −79 −2 3 −3 2 −11 7 −1 3 10 −8 −5 162 −53 −3 37 −91 −14 302 3 19 (a) $5000; (b) $8000 23. 24. (a) 512 in3 ; (b) 588 in3 9 1.2 Order of Operations Learning Objectives • Evaluate algebraic expressions with grouping symbols. • Evaluate algebraic expressions with fraction bars. • Evaluate algebraic expressions with a graphing calculator. Introduction Look at and evaluate the following expression: 2 + 4 × 7 − 1 =? How many different ways can we interpret this problem, and how many different answers could someone possibly find for it? The simplest way to evaluate the expression is simply to start at the left and work your way across, keeping track of the total as you go: 2+4=6 6 × 7 = 42 42 − 1 = 41 If you enter the expression into a non-scientific, non-graphing calculator you will probably get 41 as the answer. If, on the other hand, you were to enter the expression into a scientific calculator or a graphing calculator you would probably get 29 as an answer. In mathematics, the order in which we perform the various operations (such as adding, multiplying, etc.) is important. In the expression above, the operation of multiplication takes precedence over addition so we evaluate it first. Let’s re-write the expression, but put the multiplication in brackets to indicate that it is to be evaluated first. 2 + (4 × 7) − 1 =? So we fi...
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