Table Of Content

CK-12 F OUNDATION CK-12 Algebra I - Second Edition Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) Gloag Gloag Rawley To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook mate- rials 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®. 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Printed: August 2, 2011 Authors Andrew Gloag, Anne Gloag, Eve Rawley Editor Annamaria Farbizio Source Anne Gloag i www.ck12.org Contents 1 Equations and Functions 1 1.1 Variable Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Patterns and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.5 Functions as Rules and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6 Functions as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.7 Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 1.8 Problem-Solving Strategies: Make a Table and Look for a Pattern . . . . . . . . . . . . . . 52 2 Real Numbers 63 2.1 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 Adding and Subtracting Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 Multiplying and Dividing Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.5 Square Roots and Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.6 Problem-Solving Strategies: Guess and Check, Work Backward . . . . . . . . . . . . . . . . 95 3 Equations of Lines 102 3.1 One-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.3 Multi-Step Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.4 Equations with Variables on Both Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.5 Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.6 Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4 Graphs of Equations and Functions 138 4.1 The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.2 Graphs of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 www.ck12.org ii 4.3 Graphing Using Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4 Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.5 Graphs Using Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.6 Direct Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.7 Linear Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.8 Problem-Solving Strategies - Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5 Writing Linear Equations 197 5.1 Forms of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.2 Equations of Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 210 5.3 Fitting a Line to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 5.4 Predicting with Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 6 Linear Inequalities 237 6.1 Solving Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 6.2 Using Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.3 Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 6.4 Absolute Value Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.5 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7 Solving Systems of Equations and Inequalities 274 7.1 Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.2 Solving Linear Systems by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 7.3 Solving Linear Systems by Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 7.4 Special Types of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.5 Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 8 Exponential Functions 324 8.1 Exponent Properties Involving Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.2 Exponent Properties Involving Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.3 Zero, Negative, and Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.4 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 8.5 Geometric Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.6 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 8.7 Applications of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 9 Polynomials 368 9.1 Addition and Subtraction of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 iii www.ck12.org 9.2 Multiplication of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 9.3 Special Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 9.4 Polynomial Equations in Factored Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 9.5 Factoring Quadratic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 9.6 Factoring Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 9.7 Factoring Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 10 Quadratic Equations and Quadratic Functions 417 10.1 Graphs of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 10.2 Quadratic Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 10.3 Quadratic Equations by Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 10.4 Solving Quadratic Equations by Completing the Square . . . . . . . . . . . . . . . . . . . . 446 10.5 Solving Quadratic Equations by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . 455 10.6 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 10.7 Linear, Exponential and Quadratic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 11 Algebra and Geometry Connections 486 11.1 Graphs of Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 11.2 Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 11.3 Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506 11.4 The Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 11.5 Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 12 Rational Equations and Functions 528 12.1 Inverse Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 12.2 Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 12.3 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 12.4 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 12.5 Multiplying and Dividing Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 553 12.6 Adding and Subtracting Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 557 12.7 Solutions of Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 www.ck12.org iv Chapter 1 Equations and Functions 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 betweentwonumberswithoutneedingtoknowwhattheactualnumbersare. Thesameletterscanrepresent 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. 1 www.ck12.org 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: 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 www.ck12.org 2 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 7 −11y+2. y 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 = h(a+b). Find the area of a trapezoid with bases 2 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 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: 3 www.ck12.org 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. Foramoredetailedreviewofexponentsandtheirproperties,checkoutthevideoathttp://www.mathvids. com/lesson/mathhelp/863-exponents---basics. Example 5 The area of a circle is given by the formula A = (cid:25)r2. Find the area of a circle with radius r = 17 inches. Substitute values into the equation. A = (cid:25)r2 Substitute 17 for r: A = (cid:25)(17)2 (cid:25)·17·17 ≈ 907:9202::: Round to 2 decimal places. The area is approximately 907.92 square inches. Example 6 Find the value of x2y3 , for x = 2 and y = −4. x3+y2 Substitute the values of x and y in the following. x2y3 (2)2(−4)3 = Substitute 2 for x and −4 for y: x3+y2 (2)3+(−4)2 4(−64) −256 −32 = = Evaluate expressions: (2)2 = (2)(2) = 4 and 8+16 24 3 (2)3 = (2)(2)(2) = 8: (−4)2 = (−4)(−4) = 16 and (−4)3 = (−4)(−4)(−4) = −64: Example 7 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 www.ck12.org 4