CK-12-Algebra-I---Second-Edition_b_v4_djd_s1 - CK-12 Algebra I Second Edition Eve Rawley Anne Gloag Andrew Gloag Eve Rawley(EveR Andrew

CK-12-Algebra-I---Second-Edition_b_v4_djd_s1 - CK-12...

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Unformatted text preview: CK-12 Algebra I - Second Edition Eve Rawley Anne Gloag Andrew Gloag Eve Rawley, (EveR) Andrew Gloag, (AndrewG) Anne Gloag, (AnneG) Say Thanks to the Authors Click (No sign in required) To access a customizable version of this book, as well as other interactive content, visit 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, web-based 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, powered through the FlexBook Platform®. AUTHORS Eve Rawley Anne Gloag Andrew Gloag Eve Rawley, (EveR) Andrew Gloag, (AndrewG) Anne Gloag, (AnneG) EDITOR Annamaria Farbizio SOURCE Anne Gloag, (AnneG) Copyright © 2013 CK-12 Foundation, The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link (placed in a visible location) in addition to the following terms. 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 ( ), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at . Printed: March 29, 2013 iii Contents Contents 1 2 3 4 5 iv Equations and Functions 1.1 Variable Expressions . . . . . . . . . . . . . . . . . . . . . . . 1.2 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Patterns and Equations . . . . . . . . . . . . . . . . . . . . . . 1.4 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . 1.5 Functions as Rules and Tables . . . . . . . . . . . . . . . . . . . 1.6 Functions as Graphs . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . 1.8 Problem-Solving Strategies: Make a Table and Look for a Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 8 17 26 34 42 56 61 Real Numbers 2.1 Integers and Rational Numbers . . . . . . . . . . . . . . . . . 2.2 Adding and Subtracting Rational Numbers . . . . . . . . . . . 2.3 Multiplying and Dividing Rational Numbers . . . . . . . . . . 2.4 The Distributive Property . . . . . . . . . . . . . . . . . . . . 2.5 Square Roots and Real Numbers . . . . . . . . . . . . . . . . 2.6 Problem-Solving Strategies: Guess and Check, Work Backward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 73 81 89 99 104 110 Equations of Lines 3.1 One-Step Equations . . . . . . . . . . . 3.2 Two-Step Equations . . . . . . . . . . . 3.3 Multi-Step Equations . . . . . . . . . . 3.4 Equations with Variables on Both Sides 3.5 Ratios and Proportions . . . . . . . . . 3.6 Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 117 123 130 136 141 148 Graphs of Equations and Functions 4.1 The Coordinate Plane . . . . . . . . 4.2 Graphs of Linear Equations . . . . . 4.3 Graphing Using Intercepts . . . . . 4.4 Slope and Rate of Change . . . . . . 4.5 Graphs Using Slope-Intercept Form . 4.6 Direct Variation Models . . . . . . . 4.7 Linear Function Graphs . . . . . . . 4.8 Problem-Solving Strategies - Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 157 168 176 186 195 203 210 216 . . . . 223 224 238 247 257 . . . . . . . . . . . . . . . . Writing Linear Equations 5.1 Forms of Linear Equations . . . . . . . . . . 5.2 Equations of Parallel and Perpendicular Lines 5.3 Fitting a Line to Data . . . . . . . . . . . . . 5.4 Predicting with Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 7 8 9 Contents Linear Inequalities 6.1 Solving Inequalities . . . . . . . . . . . . 6.2 Using Inequalities . . . . . . . . . . . . . 6.3 Compound Inequalities . . . . . . . . . . 6.4 Absolute Value Equations and Inequalities 6.5 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 267 275 279 288 300 Solving Systems of Equations and Inequalities 7.1 Linear Systems by Graphing . . . . . . 7.2 Solving Linear Systems by Substitution 7.3 Solving Linear Systems by Elimination . 7.4 Special Types of Linear Systems . . . . 7.5 Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 310 320 328 340 349 Exponential Functions 8.1 Exponent Properties Involving Products . 8.2 Exponent Properties Involving Quotients . 8.3 Zero, Negative, and Fractional Exponents 8.4 Scientific Notation . . . . . . . . . . . . . 8.5 Geometric Sequences . . . . . . . . . . . 8.6 Exponential Functions . . . . . . . . . . . 8.7 Applications of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 365 371 376 383 391 397 405 . . . . . . . 415 416 425 433 439 446 454 463 Polynomials 9.1 Addition and Subtraction of Polynomials 9.2 Multiplication of Polynomials . . . . . . 9.3 Special Products of Polynomials . . . . 9.4 Polynomial Equations in Factored Form 9.5 Factoring Quadratic Expressions . . . . 9.6 Factoring Special Products . . . . . . . 9.7 Factoring Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Quadratic Equations and Quadratic Functions 10.1 Graphs of Quadratic Functions . . . . . . . . . . . . . 10.2 Quadratic Equations by Graphing . . . . . . . . . . . . 10.3 Quadratic Equations by Square Roots . . . . . . . . . . 10.4 Solving Quadratic Equations by Completing the Square 10.5 Solving Quadratic Equations by the Quadratic Formula 10.6 The Discriminant . . . . . . . . . . . . . . . . . . . . 10.7 Linear, Exponential and Quadratic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 474 488 497 505 516 526 531 11 Algebra and Geometry Connections 11.1 Graphs of Square Root Functions . . . . . . 11.2 Radical Expressions . . . . . . . . . . . . . 11.3 Radical Equations . . . . . . . . . . . . . . 11.4 The Pythagorean Theorem and Its Converse 11.5 Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 549 560 571 578 588 12 Rational Equations and Functions 12.1 Inverse Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 597 603 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Contents 12.4 12.5 12.6 12.7 Rational Expressions . . . . . . . . . . . . . . Multiplying and Dividing Rational Expressions Adding and Subtracting Rational Expressions . Solutions of Rational Equations . . . . . . . . . 13 Probability and Statistics 13.1 Theoretical and Experimental Probability . . . 13.2 Probability and Permutations . . . . . . . . . 13.3 Probability and Combinations . . . . . . . . . 13.4 Probability of Compound Events . . . . . . . 13.5 Measures of Central Tendency and Dispersion 13.6 Stem-and-Leaf Plots and Histograms . . . . . 13.7 Box-and-Whisker Plots . . . . . . . . . . . . 13.8 Surveys and Samples . . . . . . . . . . . . . vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 625 630 638 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 645 652 658 664 670 678 689 696 Chapter 1. Equations and Functions C HAPTER 1 Equations and Functions Chapter Outline 1.1 VARIABLE E XPRESSIONS 1.2 O RDER OF O PERATIONS 1.3 PATTERNS AND E QUATIONS 1.4 E QUATIONS AND I NEQUALITIES 1.5 F UNCTIONS AS R ULES AND TABLES 1.6 F UNCTIONS AS G RAPHS 1.7 P ROBLEM -S OLVING P LAN 1.8 P ROBLEM -S OLVING S TRATEGIES : M AKE A TABLE AND L OOK FOR A PATTERN 1 1.1. Variable Expressions 1.1 Variable Expressions Learning Objectives • Evaluate algebraic expressions. • Evaluate algebraic expressions with exponents. Introduction - The Language of Algebra No one likes doing the same problem over and over again—that’s why mathematicians invented algebra. Algebra takes the basic principles of math and makes them more general, so we can solve a problem once and then use that solution to solve a group of similar problems. In arithmetic, you’ve dealt with numbers and their arithmetical operations (such as +, −, ×, ÷). In algebra, we use symbols called variables (which are usually letters, such as x, y, a, b, c, . . .) to represent numbers and sometimes processes. For example, we might use the letter x to represent some number we don’t know yet, which we might need to figure out in the course of a problem. Or we might use two letters, like x and y, to show a relationship between two numbers without needing to know what the actual numbers are. The same letters can represent a wide range of possible numbers, and the same letter may represent completely different numbers when used in two different problems. Using variables offers advantages over solving each problem “from scratch.” With variables, we can: • Formulate arithmetical laws such as a + b = b + a for all real numbers a and b. • Refer to “unknown” numbers. For instance: find a number x such that 3x + 1 = 10. • Write more compactly 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 below. To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths in numbers, because we can use variables like l and w to represent the unknown length and width. If we start at the top left and work clockwise, and if we use the letter P to represent the perimeter, then we can say: 2 Chapter 1. Equations and Functions P = l +w+l +w We are adding 2 l’s and 2 w’s, so we can say that: 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 also write: P = 2l + 2w Area is length multiplied by width. In algebraic terms we get: A = l × w → A = l · w → A = lw Note: 2l + 2w by itself is an example of a variable expression; P = 2l + 2w is an example of an equation. The main difference between expressions and equations is the presence of an equals sign (=). In the above example, we found the simplest possible ways to express the perimeter and area of a rectangle when we don’t yet know what its length and width actually are. Now, when we encounter a rectangle whose dimensions we do know, we can simply substitute (or plug in) those values in the above equations. In this chapter, we will encounter many expressions that we can evaluate by plugging in values for the variables involved. Evaluate Algebraic Expressions When we are given an algebraic expression, one of the most common things we might 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, we substitute 12 for x in the given expression. Every time we see x, we replace it with 12. 2x − 7 = 2(12) − 7 = 24 − 7 = 17 Note: At this stage of the problem, we place the substituted value in parentheses. We do this to make the written-out problem easier to follow, and to avoid mistakes. (If we didn’t use parentheses and also forgot to add a multiplication sign, we would end up turning 2x into 212 instead of 2 times 12!) Example 3 Let y = −2. Find the value of 7y − 11y + 2. 3 1.1. Variable Expressions 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 4 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 and 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: h A = (a + b) Substitute 10 for a, 15 for b, and 8 for h. 2 8 8 A = (10 + 15) Evaluate piece by piece. 10 + 15 = 25; = 4. 2 2 A = 4(25) = 100 Solution: The area of the trapezoid is 100 square centimeters. 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 4 Chapter 1. Equations and Functions 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 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. For a more detailed review of exponents and their properties, check out the video at on/mathhelp/863-exponents—basics. Example 5 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. A = πr2 Substitute 17 for r. 2 A = π(17) π · 17 · 17 ≈ 907.9202 . . . Round to 2 decimal places. The area is approximately 907.92 square inches. Example 6 Find the value of x 2 y3 , x3 +y2 for x = 2 and y = −4. Substitute the values of x and y in the following. x 2 y3 (2)2 (−4)3 = x3 + y2 (2)3 + (−4)2 4(−64) −256 −32 = = 8 + 16 24 3 Substitute 2 for x and − 4 for y. Evaluate expressions: (2)2 = (2)(2) = 4 and (2)3 = (2)(2)(2) = 8. (−4)2 = (−4)(−4) = 16 and (−4)3 = (−4)(−4)(−4) = −64. Example 7 5 1.1. Variable Expressions The height (h) of a ball in flight is given by the formula h = −32t 2 + 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 = −32t 2 + 60t + 20 = −32(2)2 + 60(2) + 20 Substitute 2 for t. = −32(4) + 60(2) + 20 = 12 The height of the ball is 12 feet. Review Questions 1. Write the following in a more condensed form by leaving out a multiplication symbol. a. 2 × 11x b. 1.35 · y c. 3 × 41 d. 14 · z 2. Evaluate the following expressions for a = −3, b = 2, c = 5, and d = −4. a. 2a + 3b b. 4c + d c. 5ac − 2b 2a d. c−d e. 3b d a−4b f. 3c+2d 1 g. a+b h. ab cd 3. Evaluate the following expressions for x = −1, y = 2, z = −3, and w = 4. a. b. c. d. e. f. g. h. 8x3 5x2 6z3 3z2 − 5w2 x2 − y2 z3 +w3 z3 −w3 2x3 − 3x2 + 5x − 4 4w3 + 3w2 − w + 2 3 + z12 4. 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? c. What is the cost of producing 2500 remote controls? 5. 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. 6 Chapter 1. Equations and Functions a. What is the volume when x = 2? b. What is the volume when x = 3? 7 1.2. Order of Operations 1.2 Order of Operations Learning Objectives • Evaluate algebraic expressions with grouping symbols. • Evaluate algebraic expressions with fraction bars. • Evaluate algebraic expressions using 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: 2+4×7−1 = 6×7−1 = 42 − 1 = 41 This is t...
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