Equations, Inequalities, and Problem Solving

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1 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page 77 Equations, Inequalities, and Problem Solving Deborah Elias EVENT COORDINATOR Houston, Texas As an event planner, I am constantly using math. Calculations range from figuring the tax and gratuity percentage to add to the final bill to finding dimensions of linens to fit a table properly. For every client, I also determine a budget with income and expenses. AN APPLICATION Event promoters use the formula p = 1.x s to determine a ticket price p for an event with x dollars of expenses and s anticipated ticket sales. Grand Events expects expenses for an upcoming concert to be $0,000 and anticipates selling 000 tickets. What should the ticket price be? Source: The Indianapolis Star, /7/0 This problem appears as Example 1 in Section.. CHAPTER.1 Solving Equations. Using the Principles Together. Formulas. Applications with Percent. Problem Solving.6 Solving Inequalities CONNECTING THE CONCEPTS.7 Solving Applications with Inequalities STUDY SUMMARY REVIEW EXERCISES CHAPTER TEST CUMULATIVE REVIEW 77

2 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page 7 7 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Solving equations and inequalities is a recurring theme in much of mathematics. In this chapter, we will study some of the principles used to solve equations and inequalities. We will then use equations and inequalities to solve applied problems..1 Solving Equations Equations and Solutions a The Addition Principle a The Multiplication Principle a Selecting the Correct Approach Solving equations is essential for problem solving in algebra. In this section, we study two of the most important principles used for this task. Equations and Solutions We have already seen that an equation is a number sentence stating that the expressions on either side of the equals sign represent the same number. Some equations, like + = or x + 6 = 1x +, are always true and some, like + = 6 or x + = x +, are never true. In this text, we will concentrate on equations like x + 6 = 1 or 7x = 11 that are sometimes true, depending on the replacement value for the variable. Solution of an Equation Any replacement for the variable that makes an equation true is called a solution of the equation. To solve an equation means to find all of its solutions. To determine whether a number is a solution, we substitute that number for the variable throughout the equation. If the values on both sides of the equals sign are the same, then the number that was substituted is a solution. EXAMPLE 1 Determine whether 7 is a solution of x + 6 = 1. SOLUTION We have x + 6 = =? 1 Writing the equation Substituting 7 for x 1 = 1 is a true statement. Since the left-hand side and the right-hand side are the same, 7 is a solution. CAUTION! Note that in Example 1, the solution is 7, not 1.

3 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page 7.1 SOLVING EQUATIONS 7 EXAMPLE Determine whether is a solution of 16x = 117. SOLUTION We have 00 16x = 117 Writing the equation 16a b 117 Substituting for x FALSE The statement 10 = 117 is false. Since the left-hand side and the right-hand side differ, is not a solution. The Addition Principle Consider the equation x = =? 117 We can easily see that the solution of this equation is 7. Replacing x with 7, we get 7 = 7, which is true. Now consider the equation x + 6 = 1. In Example 1, we found that the solution of x + 6 = 1 is also 7. Although the solution of x = 7 may seem more obvious, because x + 6 = 1 and x = 7 have identical solutions, the equations are said to be equivalent. STUDENT NOTES Be sure to remember the difference between an expression and an equation. For example, a - 10 and (a - ) are equivalent expressions because they represent the same value for all replacements for a. The equations a = 10 and a = are equivalent because they have the same solution,. Equivalent Equations Equations with the same solutions are called equivalent equations. There are principles that enable us to begin with one equation and end up with an equivalent equation, like x = 7, for which the solution is obvious. One such principle concerns addition. The equation a = b says that a and b stand for the same number. Suppose this is true, and some number c is added to a. We get the same result if we add c to b, because a and b are the same number. The Addition Principle For any real numbers a, b, and c, a = b is equivalent to a + c = b + c. To visualize the addition principle, consider a balance similar to one a jeweler might use. When the two sides of a balance hold equal weight, the balance is level. If weight is then added or removed, equally, on both sides, the balance will remain level. a b a c b c

4 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page 0 0 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING When using the addition principle, we often say that we add the same number to both sides of an equation. We can also subtract the same number from both sides, since subtraction can be regarded as the addition of an opposite. EXAMPLE Solve: x + = -7. SOLUTION We can add any number we like to both sides. Since - is the opposite, or additive inverse, of, we add - to each side: x + = -7 x + - = -7 - Using the addition principle: adding - to both sides or subtracting from both sides x + 0 = -1 Simplifying; x + - = x = x + 0 x = -1. Using the identity property of 0 The equation x = -1 is equivalent to the equation x + = -7 by the addition principle, so the solution of x = -1 is the solution of x + = -7. It is obvious that the solution of x = -1 is the number -1. To check the answer in the original equation, we substitute. 0 0x + = =? -7 The solution of the original equation is = -7 is true. 11 In Example, note that because we added the opposite, or additive inverse, of, the left side of the equation simplified to x plus the additive identity, 0, or simply x. These steps effectively replaced the on the left with a 0. To solve x + a = b for x, we add -a to (or subtract a from) both sides. STUDENT NOTES EXAMPLE We can also think of undoing operations in order to isolate a variable. In Example, we began with y -. on the right side. To undo the subtraction, we add.. Solve: -6. = y -.. SOLUTION The variable is on the right side this time. We can isolate y by adding. to each side: -6. = y = y The solution is = y. -6. = y =? -6. y -. can be regarded as y Using the addition principle: Adding. to both sides eliminates -. on the right side. y = y = y + 0 = y -6. = -6. is true. 1 Note that the equations a = b and b = a have the same meaning. Thus, - 6. = y -. could have been rewritten as y -. = - 6..

5 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page 1.1 SOLVING EQUATIONS 1 The Multiplication Principle A second principle for solving equations concerns multiplying. Suppose a and b are equal. If a and b are multiplied by some number c, then ac and bc will also be equal. The Multiplication Principle For any real numbers a, b, and c, with c Z 0, a = b is equivalent to a # c = b # c. EXAMPLE Solve: x = 10. SOLUTION We can multiply both sides by any nonzero number we like. Since is the reciprocal of we decide to multiply both sides by, # x = 10 x = # 10 1 # x = x =. Using the multiplication principle: Multiplying both sides by eliminates the on the left. Simplifying Using the identity property of 1 : 0 10 =? 10 The solution is. x = 10 # 10 Think of as = 10 is true. In Example, to get x alone, we multiplied by the reciprocal, or multiplicative inverse of. We then simplified the left-hand side to x times the multiplicative identity, 1, or simply x. These steps effectively replaced the on the left with 1. Because division is the same as multiplying by a reciprocal, the multiplication principle also tells us that we can divide both sides by the same nonzero number. That is, if a = b, then 1 c # a = 1 c # b and a c = b c 1provided c Z 0. STUDENT NOTES In a product like x, the multiplier is called the coefficient. When the coefficient of the variable is an integer or a decimal, it is usually easiest to solve an equation by dividing on both sides. When the coefficient is in fraction notation, it is usually easiest to multiply by the reciprocal. y EXAMPLE 6 Solve: (a) -x = ; (b) -x = ; (c). = In Example 6(a), we can think of undoing the multiplication - # x by dividing by -. SOLUTION a) In -x =, the coefficient of x is an integer, so we divide on both sides: -x - = - 1 # x = - Using the multiplication principle: Dividing both sides by - is the same as multiplying by - 1. Simplifying x = -. Using the identity property of 1

6 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING -x = -A- B The solution is. = is true. b) To solve an equation like -x =, remember that when an expression is multiplied or divided by -1, its sign is changed. Here we divide both sides by -1 to change the sign of -x: -x = -x -1 = -1 x = -. Note that -x = -1 # x. Dividing both sides by -1. (Multiplying by -1 would also work. Note that the reciprocal of -1 is -1.) -x x Note that is the same as x = -(-)? = = is true. The solution is -. y c) To solve an equation like we rewrite the left-hand side as y and then =, use the multiplication principle, multiplying by the reciprocal of : y = # y = =? - # # y = # 1y = # # # # y = 1. y Rewriting as # y Multiplying both sides by Removing a factor equal to 1: # # = 1 y # 1 = =? = is true. The solution is 1. Selecting the Correct Approach It is important that you be able to determine which principle should be used to solve a particular equation.

7 M0_BITT717_0_C0_01 pp.qxd 10/7/0 : PM Page.1 SOLVING EQUATIONS STUDY SKILLS Seeking Help? EXAMPLE 7 Solve: (a) - + x = ; (b) 1.6 = t. A variety of resources are available to help make studying easier and more enjoyable. Textbook supplements. See the preface for a description of the supplements for this textbook: the Student s Solutions Manual, a complete set of videos on DVD, MathXL tutorial exercises on CD, and complete online courses in MathXL and MyMathLab. Your college or university. Your own college or university probably has resources to enhance your math learning: a learning lab or tutoring center, study skills workshops or group tutoring sessions tailored for the course you are taking, or a bulletin board or network where you can locate the names of experienced private tutors. Your instructor. Find out your instructor s office hours and make it a point to visit when you need additional help. Many instructors also welcome student . SOLUTION a) To undo addition of - we subtract - from both sides. Subtracting -, is the same as adding. The solution is - # = - 6 Removing a factor equal to 1: = 1 b) To undo multiplication by, we either divide both sides by or multiply both 1 sides by - + x + = + : 1.6 = t x = = t. = t. x = + x = # + # x = x = x = # 6 # =? = t =? 1.6 The solution is.. Using the multiplication principle Simplifying Using the addition principle Finding a common denominator = is true. 1.6 = 1.6 is true. S and 67.1 EXERCISE SET For Extra Help i Concept Reinforcement For each of Exercises 1 6, match the statement with the most appropriate choice from the column on the right. 1. The equations x + = 7 and 6x =. The expressions 1x - and x - 6. A replacement that makes an equation true. The role of in ab. The principle used to solve # x = - 6. The principle used to solve + x = - a) Coefficient b) Equivalent expressions c) Equivalent equations d) The multiplication principle e) The addition principle f) Solution