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CHAPTER 1 Introduction to Algebra 1-1 Variables DEFINITION Variable – is a symbol used to represent one or more numbers. The numbers are called values of the variable. DEFINITION Variable Expression – an expression that contains a variable. Examples: 4x, 10y, -1/2z DEFINITION Numerical Expression – an expression that names a particular number Examples: 4.50 x 4, 6 + 2, 10-3 DEFINITION Value of the Expression – the number named by an expression Examples: 4.50 x 4 = 18 6+2=8 10-3 = 7 SUBSTITUTION PRINCIPLE An expression may be replaced by another expression that has the same value. Example: (42 ÷ 6) + 8 7+8 15 Section 1-2 Grouping Symbols Order of Operations Parenthesis Exponents Multiplication Division Addition Subtraction DEFINITION Grouping symbol – is a device, such as a pair of parentheses, used to enclose an expression that should be simplified first. Examples: ( ), { }, [ ], _____ Example 1 Simplify: a) 6(5 – 3) 12 b) 6(5) – 3 27 Example 2 Simplify: 12 + 4 15 – 7 2 Example 3 Simplify: 18 – [52 ÷ (7 + 6)] 14 Example 4 Simplify: 19 – 7 + 12 ∙ 2 ÷ 8 15 Example 5 Evaluate 4x + 5y 3x – y when x = 3 and y = 8. Section 1-3 Equations DEFINITION Equation – is formed by placing an equals sign between two numerical or variable expressions, called the sides of the equation. Examples: 11-7 = 4, 5x -1 = 9 DEFINITION Open sentences – an equation or inequality containing a variable. Examples: y + 1= 1 + y 5x -1 = 9 DEFINITION Domain – the given set of numbers that a variable may represent Example: 5x – 1 = 9 The domain of x is {1,2,3} DEFINITION Solution Set – the set of all solutions of an open sentence. Finding the solution set is called solving the sentence. Examples: y(4 - y) = 3 when y{0,1,2,3} y {1,3} Section 1-4 Translating Words into Symbols Addition - Phrases • The sum of 8 and x • A number increased by 7 • 5 more than a number Subtraction - Phrases • The difference between a number and 4 • A number decreased by 8 • 5 less than a number • 6 minus a number Multiplication - Phrases • The product of 4 and a number • Seven times a number • One third of a number Division - Phrases • The quotient of a number and 8 • A number divided by 10 Section 1-5 Translating Sentences into Equations EXAMPLES Twice the sum of a number and four is 10 2(n +4) = 10 EXAMPLES When a number is multiplied by four and the result decreased by six, the final result is 10. 4n - 6 = 10 EXAMPLES Three less than a number is 12. x – 3 = 12 EXAMPLES The quotient of a number and 4 is 8. b/4 = 8 EXAMPLES Write an equation to represent the given information. The distance traveled in 3 hours of driving was 240 km. Section 1-6 Translating Problems into Equations PROCEDURE • Read the problem carefully • Choose a variable to represent the unknowns • Reread the problem and write an equation. EXAMPLES Translate the problem into an equation. (1) Marta has twice as much money as Heidi. (2) Together they have $36. How much money does each have? Translation Let h = Heidi’s amount Then 2h = Marta’s amount h + 2h = 36 EXAMPLES Translate the problem into an equation. (1) A wooden rod 60 in. long is sawed into two pieces. (2) One piece is 4 in. longer than the other. What are the lengths of the pieces? Translation Let x = the shorter length Then x + 4 = longer length x + (x + 4) = 60 EXAMPLES Translate the problem into an equation. (1) The area of a rectangle is 102 cm2. (2) The length of the rectangle is 6 cm. Find the width of the rectangle? Translation Let w = width of rectangle Then 6 = length of rectangle 6w = 102 Section 1-7 A Problem Solving Plan SOLVING A WORD PROBLEM 1. Read the problems carefully. Decide what unknown numbers are asked for and what facts are known. Making a sketch may help SOLVING A WORD PROBLEM 2. Choose a variable and use it with the given facts to represent the unknowns described in the problem. SOLVING A WORD PROBLEM 2. Reread the problem and write an equation that represents relationships among the numbers in the problem. SOLVING A WORD PROBLEM 4. Solve the equation and find the unknowns asked for. 5. Check your results with the words of the problem. Give the answer. EXAMPLES Two numbers have a sum of 44. The larger number is 8 more than the smaller. Find the numbers. Solution n + (n + 8) = 44 2n + 8 = 44 2n = 36 n = 18 EXAMPLES • Jason has one and a half times as many books as Ramone. Together they have 45 books. How many books does each boy have? Translation Let r = number of Ramone’s books Then 1.5r = number of Jason’s books r + 1.5r = 45 Solution r + 1.5r = 45 2.5r = 45 r = 18 Examples Phillip has $23 more than Kevin. Together they have $187. How much does each have? Section 1-8 Number Lines NATURAL NUMBERS set of counting numbers {1, 2, 3, 4, 5, 6, 7, 8…} WHOLE NUMBERS – set of counting numbers plus zero {0, 1, 2, 3, 4, 5, 6, 7, 8…} INTEGERS – set of the whole numbers plus their opposites {…, -3, -2, -1, 0, 1, 2, 3, …} RATIONAL NUMBERS numbers that can be expressed as a ratio of two integers a and b and includes fractions, repeating decimals, and terminating decimals EXAMPLES OF RATIONAL NUMBERS ½, ¾, ¼, - ½, -¾, -¼, .05 .76, .333…, .666…, etc . IRRATIONAL NUMBERS numbers that cannot be expressed as a ratio of two integers a and b and can still be designated on a number line Inequality Symbols Are used to show the order of two real numbers > means “is greater than” < means “is less than” Section 1-9 Opposites and Absolute Values OPPOSITES A pair of numbers differing in sign only {-4, 4} , {10, -10}, {½, -½} RULES 1. If a is positive, then –a is negative 2. If a is negative, then –a is positive. RULES 3. If a = 0, then –a = 0 4.The opposite of –a is a; that is, -(-a) = a ABSOLUTE VALUES The absolute value of a number a is denoted by |a|, and it may be thought of as the distance between the graph of the number and the origin on a number line. EXAMPLES |8| = 8 |-8| = 8 |0|= 0 |-4| + |7|= 11 The End