1 Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a $0 setup fee plus $ per person. The cost of Hotel Pharmacy s end-of-season softball party cannot exceed $0. How many people can attend the party? This problem appears as Example in Section.7. Connecting the Concepts. Formulas. Applications with Percent. Problem Solving.6 Solving Inequalities.7 Solving Applications with Inequalities Summary and Review Test Most people might not associate math with barbeque, but in fact I use math in everything from ordering ingredients for my secret sauce to keeping the books for my business. CURTIS TUFF Chef and Owner, Curtis Barbeque Putney, VT
2 70 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. Solving Equations. Equations and Solutions The Addition Principle The Multiplication Principle 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 x, are always true and some, like 6 or x x, are never true. In this text, we will concentrate on equations like x 6 or 7x 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 Determine whether 7 is a solution of x 6. Solution We have x 6 7 6? Writing the equation Substituting 7 for x Note that 7, not, is the solution. Since the left-hand and the right-hand sides are the same, 7 is a solution.
3 . SOLVING EQUATIONS 7 Example Determine whether is a solution of 7x. Solution We have 7x 7? Writing the equation Substituting for x FALSE The statement is false. Since the left-hand and the right-hand sides differ, is not a solution. The Addition Principle Consider the equation x 7. 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. In Example, we found that the solution of x 6 is also 7. Although the solution of x 7 may seem more obvious, the equations x 6 and x 7 are equivalent. 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. (See the figure on the following page.) When the two sides of the balance hold quantities of equal weight, the balance is level. If weight is added or removed, equally, on both sides, the balance will remain level.
4 7 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING a = b a + c = b + c 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 ; x x x 0 x. Using the identity property of 0 It is obvious that the solution of x is the number. To check the answer in the original equation, we substitute. x 7? The solution of the original equation is. 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. When solving x a b for x, we simply add a to (or subtract a from) both sides. Example Solve: 6. y 8.. Solution The variable is on the right side this time. We can isolate y by adding 8. to each side: 6. y y y. This can be regarded as 6. y 8.. Using the addition principle: Adding 8. to both sides eliminates 8. on the right side. y y y 0 y
5 . SOLVING EQUATIONS 7 6. y 8. 6.? The solution is.. Note that the equations a b and b a have the same meaning. Thus, 6. y 8. could have been rewritten as y Example Solve: x. Solution We have x x x Adding to both sides Multiplying by to obtain a common denominator The check is left to the student. The solution is 6. 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 0, a b is equivalent to a c b c. Example 6 Solve: x 0. Solution We can multiply both sides by any nonzero number we like. Since is the reciprocal of we multiply each side by x 0 x 0 x 8 x 8., : Using the multiplication principle: Multiplying both sides by eliminates the on the left. Using the identity property of
6 7 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING x 0 The solution is 8. 8? In Example 6, 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,, or simply x. These steps effectively replaced the on the left with. 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, a if a b, then and (provided c 0). c b c a c b c 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 easier to multiply by the reciprocal. Example 7 Solve: (a) x ; (b).6 t; (c) x ; (d) Solution a) x x x x y 8. Using the multiplication principle: Dividing both sides by is the same as multiplying by. Using the identity property of b) The solution is..6 t.6 t. t. t x? Dividing both sides by or multiplying both sides by.6 t.6?..6.6 The solution is..
7 . SOLVING EQUATIONS 7 Study Tip This text is accompanied by a complete set of videotapes in which the authors and other instructors present material and concepts from every section of the text. These videos are available on CD-ROM and in most campus math labs. When missing a class is unavoidable, try to view the appropriate video lecture for that section. These videotapes can also be used as an aid whenever difficulty arises. c) To solve an equation like x, remember that when an expression is multiplied or divided by, its sign is changed. Here we multiply both sides by to change the sign of x : x x x. x? The solution is. Multiplying both sides by (Dividing by would also work) Note that x is the same as x. d) To solve an equation like 8, we rewrite the left-hand side as y and then use the multiplication principle: y 8 y 8 y 8 y y. y Rewriting y as y Multiplying both sides by Removing a factor equal to : 8 y 8 The solution is.? 8 8 Exercise Set. FOR EXTRA HELP Digital Video Tutor CD InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape Solve using the addition principle. Don t forget to check!. x 8. x 8. t. y. y t 7. x y. x 6 0. x 8
8 76 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Aha! Aha!. y 6 8. x. t 6. t 8. 7 y 6. z 7. t 8. 6 y. 7 r 0. t x x 6. x. y z y 8 7. m y y x 0.6 Solve using the multiplication principle. Don t forget to check!. x x. t 6. 6x x t 8 7. x x 00. t r 68. 7x 7. x 6. x 7 6. x 0 7..t a 0. 8 a y x x 6 0 t x x. 6. r r x Solve. The icon indicates an exercise designed to give practice using a calculator... t x x z y t 7... Aha! 6. y t 6. a 66. x x t x t x When solving an equation, how do you determine what number to add, subtract, multiply, or divide by on both sides of that equation? 7. What is the difference between equivalent expressions and equivalent equations? SKILL MAINTENANCE Simplify [.8] [.8] [.8] [.8] 6 SYNTHESIS y To solve. t, Anita adds. to both sides. Will this form an equivalent equation? Will it help solve the equation? Explain. 80. Explain why it is not necessary to state a subtraction principle: For any real numbers a, b, and c, a b is equivalent to a c b c. Some equations, like 7 or x x, have no solution and are called contradictions. Other equations, like 7 7 or x x, are true for all numbers and are called identities. Solve each of the following and if an identity or contradiction is found, state this. 8. x x x Identity 8. x x x Contradiction 8. x x x x x Identity 8. x 8 x x 0 0 Contradiction 87. x 88. x 6, Contradiction Solve for x. Assume a, c, m mx.m. 0. x a a
9 . USING THE PRINCIPLES TOGETHER 77 7cx. 6. c cx 7c a a c. a ax a 8. x 6,. If x 70 6, find x 70.,07 6. Lydia makes a calculation and gets an answer of.. On the last step, she multiplies by 0. when she should have divided by 0.. What should the correct answer be? 0 7. Are the equations x and x equivalent? Why or why not? Using the Principles Together. Applying Both Principles Combining Like Terms Clearing Fractions and Decimals CONNECTING THE CONCEPTS We have stated that most of algebra involves either simplifying expressions (by writing equivalent expressions) or solving equations (by writing equivalent equations). In Section., we used the addition and multiplication principles to produce equivalent equations, like x, from which the solution in this case, is obvious. Here in Section., we will find that more complicated equations can be solved by using both principles together and by using the commutative, associative, and distributive laws to write equivalent expressions. An important strategy for solving a new problem is to find a way to make the new problem look like a problem we already know how to solve. This is precisely the approach taken in this section. You will find that the last steps of the examples in this section are nearly identical to the steps used for solving the examples of Section.. What is new in this section appears in the early steps of each example. Before reading this section, make sure that you thoroughly understand the material in Section.. Without a solid grasp of how and when to use the addition and multiplication principles, the problems in this section will seem much more difficult than they really are. Applying Both Principles In the expression x, the variable x is multiplied by and then is added. To reverse these steps, we first subtract and then divide by. Thus, to solve x 7, we first subtract from each side and then divide both sides by.
10 78 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Example Solve: Solution First isolate the x-term. x 7. Then isolate x. We have x 7 x 7 x x x x. Using the addition principle: subtracting from both sides (adding ) Using a commutative law. Try to perform this step mentally. Using the multiplication principle: dividing both sides by multiplying by The solution is. x 7? We use the rules for order of operations: Find the product,, and then add. Multiplication by a negative number and subtraction are handled in much the same way. Example Solve: x 6 6. Solution In x 6 we multiply first and then subtract. To reverse these steps, we first add 6 and then divide by. x 6 6 x x x Adding 6 to both sides Dividing both sides by x, or 6? The solution is. x 6 6
11 . USING THE PRINCIPLES TOGETHER 7 Example Solve: t. Solution We have t t t t t t t. Subtracting from both sides Try to do these steps mentally. Try to go directly to this step. Multiplying both sides by (Dividing by would also work.) t? The solution is. As our skills improve, many of the steps can be streamlined. Example Solve: Solution 6. 7.y 8.8. We have 6. 7.y y y.8 y.8 7. y.. Subtracting 6. from both sides. We write the subtraction of 6. on the right side and remove 6. from the left side. Dividing both sides by 7.. We write the division by 7. on the right side and remove the 7. from the left side. The solution is y ? Combining Like Terms If like terms appear on the same side of an equation, we combine them and then solve. Should like terms appear on both sides of an equation, we can use the addition principle to rewrite all like terms on one side.
12 80 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Example technology connection Most graphers have a TABLE feature that enables the calculator to evaluate a variable expression for different choices of x. For example, to evaluate 6x 7x for x 0,,,, we first use Y to enter 6x 7x as y. We then use nd TBLSET to specify which x-values will be used. Using TblStart 0, Tbl, and selecting Auto twice, we can generate a table in which the value of 6x 7x is listed for values of x starting at 0 and increasing by ones. 0 6 X X 0 0 Y Solve. a) x x b) x x c) 6x 7x 0 x 7 d) x x Solution a) x x 7x x 7 Combining like terms Dividing both sides by 7 x The check is left to the student. The solution is. b) To solve x x, we must first write only variable terms on one side and only constant terms on the other. This can be done by adding to both sides, to get all constant terms on the right, and x to both sides, to get all variable terms on the left. We can add first, or x first, or do both in one step. Isolate variable terms on one side and constant terms on the other side. x x x x x x x x x x x x x Adding to both sides Adding x to both sides Combining like terms and simplifying Dividing both sides by. Create the above table on your grapher. Scroll up and down to extend the table.. Enter 0 x 7 as y. Your table should now have three columns.. For what x-value is y the same as y? Compare this with the solution of Example (c). Is this a reliable way to solve equations? Why or why not?. X Y. 0 6 X 0 0 X 0 6 X 0 Y 0 Y 7 7. ; not reliable because, depending on the choice of Tbl, it is easy to scroll past a solution without realizing it. c) x x? The solution is. 6x 7x 0 x 7 x 7 x x x 7 x x x 7 x x 6x 7x 0 x 7 6 7? The solution is. Combining like terms on both sides Adding x to both sides. This is identical to Example. Subtracting from both sides and simplifying Dividing both sides by and simplifying
13 . USING THE PRINCIPLES TOGETHER 8 d) x x x x 6 x x 7 x 7 x 7 x x 6 8x x Using the distributive law. This is now similar to part (c) above. Combining like terms on both sides Adding 7 and x to both sides. This isolates the x-terms on one side and the constant terms on the other. Dividing both sides by 8 The student can confirm that checks and is the solution. Clearing Fractions and Decimals Equations are generally easier to solve when they do not contain fractions or decimals. The multiplication principle can be used to clear fractions or decimals, as shown here. Clearing Fractions x x x 0 Clearing Decimals.x 7. 0.x 7 0. x 70 In each case, the resulting equation is equivalent to the original equation, but easier to solve. The easiest way to clear an equation of fractions is to multiply both sides of the equation by the smallest, or least, common denominator. Example 6 Solve: (a) x 6 x; (b) Solution a) The number 6 is the least common denominator, so we multiply both sides by 6. 6 x 6 6 x x 8. Multiplying both sides by 6 6 x x Caution! Be sure the distributive law is used to multiply all the terms by 6. x x 8x 8 x. Note that the fractions are cleared. Subtracting x from both sides Dividing both sides by 8 The number 8 checks and is the solution.
14 8 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING b) To solve x 8, we can multiply both sides by or divide by to undo the multiplication by on the left side. x 8 x 0 x 8 x 6 Multiplying both sides by ; and 8 0 Subtracting from both sides Dividing both sides by The student can confirm that 6 checks and is the solution. To clear an equation of decimals, we count the greatest number of decimal places in any one number. If the greatest number of decimal places is, we multiply both sides by 0; if it is, we multiply by 00; and so on. Example 7 Solve: 6. 7.y 8.8. Solution The greatest number of decimal places in any one number is two. Multiplying by 00 will clear all decimals y y y 88 70y y 8 y 8 70 Multiplying both sides by 00 Using the distributive law Subtracting 60 from both sides Combining like terms Dividing both sides by 70 y. In Example, the same solution was found without clearing decimals. Finding the same answer two ways is a good check. The solution is.. An Equation-Solving Procedure. Use the multiplication principle to clear any fractions or decimals. (This is optional, but can ease computations.). If necessary, use the distributive law to remove parentheses. Then combine like terms on each side.. Use the addition principle, as needed, to get all variable terms on one side and all constant terms on the other.. Combine like terms again, if necessary.. Multiply or divide to solve for the variable, using the multiplication principle. 6. Check all possible solutions in the original equation.
15 . USING THE PRINCIPLES TOGETHER 8 Exercise Set. FOR EXTRA HELP Digital Video Tutor CD InterAct Math Math Tutor Center MathXL MyMathLab.com Videotape Solve and check.. x 8 7. x Aha!. 8x z 7. 7t x 7. x 8. x 0. 8z 7 0. x 6. 8x. t 8. x 7 7. x 08. 7x 6. 6z x 7x 8. x x. 7 6x 0. 7x. x x 6. 6x x 00. a a 8. y 7y. 7y 8y 6. 0y y y 7.y t.t 0. x 6 0. x x 8 x 0 8. y y 7. x 6 6x. 6x 7 x. y 8 y. 6x x 0 6. y y 7. x x 7x x x 8x x 6 x x 0. x 7 x x 0. y y 6y 0 y 0. y 0 y 7y 8 y 7 Clear fractions or decimals, solve, and check.. x x x x 7. 8 x x 6 x 6. t 6 6. x 6 7. t 6t 8. m m. x x 0. y y..x.. 8.x z. 0.6z t 0.6t t 8.6t 0.t x x x y 8 y y 7 Solve and check a 8. t. x 60. x 6. m m r r b b x 66. d 7 d t t t 7t x 6 6x t t 6 7. x x x 7. c c c 7. t 8 7. x 7 7. x t x x x 6 8
16 8 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING x x 6. x x 8 0 x 8. a a a a b 0. b When an equation contains decimals, is it essential to clear the equation of decimals? Why or why not? 86. Why must the rules for the order of operations be understood before solving the equations in this section? SKILL MAINTENANCE Evaluate. 87. a, for a [.8] t, for t [.8] 8. 7x x, for x [.8] 0. t 8 t, for t [.8] 8 SYNTHESIS. What procedure would you follow to solve an equation like 0.x x? Could your procedure be streamlined? If so, how? Aha!. Dave is determined to solve the equation x by first using the multiplication principle to eliminate the. How should he proceed and why? Solve. If an equation is an identity or a contradiction (see p. 76), state this x.. 0.7x.x.0 0, or x.8 0.x x.6. x x x Contradiction 6. 0 y y 7 7. x x Identity 8. x 7 x x Contradiction. x x x x x x 00. x x x x x x 0. x x x x 0 7x x Identity x x x x 7 x 0. x COLLABORATIVE CORNER Step-by-Step Solutions Focus: Solving linear equations Time: 0 minutes Group size: In general, there is more than one correct sequence of steps for solving an equation. This makes it important that you write your steps clearly and logically so that others can follow your approach. ACTIVITY. Each group member should select a different one of the following equations and, on a fresh sheet of paper, perform the first step of the solution. x 7x 6 x 7 x x 6 x x 7 x 7 x x x. Pass the papers around so that the second and third steps of each solution are performed by the other two group members. Before writing, make sure that the previous step is correct. If a mistake is discovered, return the problem to the person who made the mistake for repairs. Continue passing the problems around until all equations have been solved.. Each group should reach a consensus on what the three solutions are and then compare their answers to those of other groups.