P.3 Linear Equations and Inequalities

Transcription

1 SECTION P. Linear Equations and Inequalities 1 P. Linear Equations and Inequalities What you ll learn about Equations Solving Equations Linear Equations in One Variable Linear Inequalities in One Variable... and why These topics provide the foundation for algebraic techniques needed throughout this tetbook. Equations An equation is a statement of equality between two epressions. Here are some properties of equality that we use to solve equations algebraically. Properties of Equality Let u, v, w, and z be real numbers, variables, or algebraic epressions. 1. Refleive u = u. Symmetric If u = v, then v = u.. Transitive If u = v, and v = w, then u = w.. Addition If u = v and w = z, then u + w = v + z.. Multiplication If u = v and w = z, then uw = vz. Solving Equations A solution of an equation in is a value of for which the equation is true. To solve an equation in means to find all values of for which the equation is true, that is, to find all solutions of the equation. EXAMPLE 1 Confirming a Solution Prove that =- is a solution of the equation = = Now try Eercise 1. Linear Equations in One Variable The most basic equation in algebra is a linear equation. DEFINITION Linear Equation in A linear equation in is one that can be written in the form a + b =, where a and b are real numbers and a. The equation z - = is linear in the variable z. The equation u - 1 = is not linear in the variable u. A linear equation in one variable has eactly one solution. We solve such an equation by transforming it into an equivalent equation whose solution is obvious. Two or more equations are equivalent if they have the same solutions. For eample, the equations z - =, z =, and z = are all equivalent. Here are operations that produce equivalent equations.

2 CHAPTER P Prerequisites Operations for Equivalent Equations An equivalent equation is obtained if one or more of the following operations are performed. Given Equivalent Operation Equation Equation 1. Combine like terms, reduce fractions, and remove grouping + = 9 = 1 symbols.. Perform the same operation on both sides. (a) Add = 7 = (b) Subtract (). = + = (c) Multiply by a nonzero constant (1/). = 1 = (d) Divide by a nonzero constant (). = 1 = The net two eamples illustrate how to use equivalent equations to solve linear equations. EXAMPLE Solving a Linear Equation Solve = +. Support the result with a calculator = = = + Distributive property Combine like terms. = Add, and subtract. =. Divide by. To support our algebraic work we can evaluate the epressions in the original equation for =.. Figure P.17 shows that each side of the original equation is equal to 1. if =.. Now try Eercise.. X (X )+(X+1) X FIGURE P.17 The top line stores the number. into the variable. (Eample ) If an equation involves fractions, find the least common denominator (LCD) of the fractions and multiply both sides by the LCD. This is sometimes referred to as clearing the equation of fractions. Eample illustrates.

3 SECTION P. Linear Equations and Inequalities Integers and Fractions Notice in Eample that = 1. Solve EXAMPLE Solving a Linear Equation Involving Fractions y - = + y. The denominators are, 1, and. The LCD of the fractions is. (See Appendi A. if necessary.) y - = + y a y - b = a + y b Multiply by the LCD. # y - = # + # y Distributive property y - = 16 + y Simplify. y = 1 + y Add. y = 1 Subtract y. y = 6 Divide by. We leave it to you to check the solution using either paper and pencil or a calculator. Now try Eercise. Linear Inequalities in One Variable We used inequalities to describe order on the number line in Section P.1. For eample, if is to the left of on the number line, or if is any real number less than, we write 6. The most basic inequality in algebra is a linear inequality. DEFINITION Linear Inequality in A linear inequality in is one that can be written in the form a + b 6, a + b, a + b 7, or a + b Ú, where a and b are real numbers and a. Direction of an Inequality Multiplying (or dividing) an inequality by a positive number preserves the direction of the inequality. Multiplying (or dividing) an in equality by a negative number reverses the direction. To solve an inequality in means to find all values of for which the inequality is true. A solution of an inequality in is a value of for which the inequality is true. The set of all solutions of an inequality is the solution set of the inequality. We solve an inequality by finding its solution set. Here is a list of properties we use to solve inequalities. Properties of Inequalities Let u, v, w, and z be real numbers, variables, or algebraic epressions, and c a real number. 1. Transitive If u 6 v and v 6 w, then u 6 w.. Addition If u 6 v, then u + w 6 v + w. If u 6 v and w 6 z, then u + w 6 v + z.. Multiplication If u 6 v and c 7, then uc 6 vc. If u 6 v and c 6, then uc 7 vc. The above properties are true if 6 is replaced by. There are similar properties for 7 and Ú.

4 CHAPTER P Prerequisites The set of solutions of a linear inequality in one variable forms an interval of real numbers. Just as with linear equations, we solve a linear inequality by transforming it into an equivalent inequality whose solutions are obvious. Two or more inequalities are equivalent if they have the same solution set. The properties of inequalities listed above describe operations that transform an inequality into an equivalent one. EXAMPLE Solving a Linear Inequality Solve ( - 1) Distributive property Combine like terms. + 7 Add Subtract. a- 1 b # - Ú a- 1 b # 7 Multiply by -1/. (The inequality reverses.) Ú-. The solution set of the inequality is the set of all real numbers greater than or equal to -.. In interval notation, the solution set is -.,. Now try Eercise 1. Because the solution set of a linear inequality is an interval of real numbers, we can display the solution set with a number line graph as illustrated in Eample. EXAMPLE Solving a Linear Inequality Involving Fractions Solve the inequality, and graph its solution set The LCD of the fractions is # a + 1 b 7 1 # a + 1 b Multiply by the LCD Simplify Subtract. 7- Subtract 6. The solution set is the interval 1-,. Its graph is shown in Figure P.1. Now try Eercise. 1 1 FIGURE P.1 The graph of the solution set of the inequality in Eample.

5 SECTION P. Linear Equations and Inequalities Sometimes two inequalities are combined in a double inequality, which is solved by isolating as the middle epression. Eample 6 illustrates this. EXAMPLE 6 Solving a Double Inequality Solve the inequality, and graph its solution set FIGURE P.19 The graph of the solution set of the double inequality in Eample Multiply by Subtract Divide by. The solution set is the set of all real numbers greater than -7 and less than or equal to. In interval notation, the solution is set 1-7,. Its graph is shown in Figure P.19. Now try Eercise 7. QUICK REVIEW P. In Eercises 1 and, simplify the epression by combining like terms y - + y z + y - + y - z - In Eercises and, use the distributive property to epand the products. Simplify the resulting epression by combining like terms y + 1y y. 1 + y y In Eercises 1, use the LCD to combine the fractions. Simplify the resulting fraction.. y + y y y y SECTION P. Eercises Eercise numbers with a gray background indicate problems that the authors have designed to be solved without a calculator. In Eercises 1, which values of are solutions of the equation? 1. + = (a) =- (b) =- 1 (c) = = (a) =-1 (b) = (c) = = (a) =- (b) = (c) =. 1-1/ = (a) =-6 (b) = (c) = 1 In Eercises 1, determine whether the equation is linear in.. - = 6. = 1/ 7. + = -. - = 9. + = = 1 In Eercises 11, solve the equation without using a calculator. 11. = 1. = t - = 1. t - 9 = 1. - = = y = 1y y - = y = 7. =

6 6 CHAPTER P Prerequisites = = z - 1z + = z z - - 1z + 1 = z - In Eercises, solve the equation. Support your answer with a calculator t + + = 6. - = - - t - = 1. t t + = 1 9. Writing to Learn Write a statement about a solution of an equation suggested by the computations in the figure. (a) X (b) / X 1. X +X 6 X +X 6. Ú y - Ú y z t y -. - y 6 + y y - 6 y - 1 Ú - y In Eercises, find the solutions of the equation or inequality that are displayed in Figure P = Writing to Learn Write a statement about a solution of an equation suggested by the computations in the figure. (a) X (b) X 7X+ 7X+ 19 X 7 X 7 1 In Eercises 1, which values of are solutions of the inequality? (a) = (b) = (c) = 6. - Ú (a) = (b) = (c) = (a) = (b) = (c) = (a) =-1 (b) = (c) = In Eercises, solve the inequality, and draw a number line graph of the solution set Ú In Eercises, solve the inequality. 1 6 X Y1 1 1 Y1 = X X FIGURE P. The second column gives values of y 1 = - for =, 1,,,,, and Writing to Learn Eplain how the second equation was obtained from the first. - = +, - 6 = Writing to Learn Eplain how the second equation was obtained from the first. - 1 = -, - 1 = Group Activity Determine whether the two equations are equivalent. (a) = 6 + 9, = + 9 (b) 6 + = + 1, + 1 = + 6. Group Activity Determine whether the two equations are equivalent. (a) + = - 7, - + =-7 (b) + = - 7, =

7 SECTION P. Linear Equations and Inequalities 7 Standardized Test Questions 6. True or False Justify your answer. 6. True or False 6. Justify your answer. In Eercises 6 6, you may use a graphing calculator to solve these problems. 6. Multiple Choice Which of the following equations is equivalent to the equation + = + 1? (A) = (B) = + (C) + = + 1 (D) + 6 = (E) = Multiple Choice Which of the following inequalities is equivalent to the inequality - 6 6? (A) 6-6 (B) 6 1 (C) 7- (D) 7 (E) Multiple Choice Which of the following is the solution to the equation =? (A) = or =-1 (B) = or = 1 (C) Only =-1 (D) Only = (E) Only = 1 6. Multiple Choice Which of the following represents an equation equivalent to the equation + 1 = - 1 that is cleared of fractions? (A) + 1 = - 1 (B) + 6 = - (C) + = - (D) + = - (E) + 6 = - Eplorations 69. Testing Inequalities on a Calculator (a) The calculator we use indicates that the statement 6 is true by returning the value 1 (for true) when 6 is entered. Try it with your calculator. (b) The calculator we use indicates that the statement 6 1 is false by returning the value (for false) when 6 1 is entered. Try it with your calculator. (c) Use your calculator to test which of these two numbers is larger: 799/, /1. (d) Use your calculator to test which of these two numbers is larger: -1/11, -1/1. (e) If your calculator returns when you enter + 1 6, what can you conclude about the value stored in? Etending the Ideas 7. Perimeter of a Rectangle The formula for the perimeter P of a rectangle is P = 1L + W. Solve this equation for W. 71. Area of a Trapezoid The formula for the area A of a trapezoid is A = 1 h1b 1 + b. Solve this equation for b Volume of a Sphere The formula for the volume V of a sphere is V = pr. Solve this equation for r. 7. Celsius and Fahrenheit The formula for Celsius temperature in terms of Fahrenheit temperature is Solve the equation for F. C = 1F -. 9