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Test Bank for Algebra for College Students, 8th Edition

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    I NSTRUCTOR ’ S R ESOURCE M ANUAL A LGEBRA FOR C OLLEGE S TUDENTS E IGHTH E DITION Robert Blitzer Miami Dade College

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    Instructor’s Resource Manual with Tests Algebra for College Students , Eighth Edition Robert Blitzer TABLE OF CONTENTS MINI-LECTURES (per section) ML-1 Chapter 1 ML-1 Chapter 2 ML-11 Chapter 3 ML-19 Chapter 4 ML-25 Chapter 5 ML-32 Chapter 6 ML-43 Chapter 7 ML-52 Chapter 8 ML-61 Chapter 9 ML-67 Chapter 10 ML-75 Chapter 11 ML-83 Chapter 12 ML-88 Mini-Lectures Answers Included at end of section ADDITIONAL EXERCISES (per section) AE-1 Chapter 1 AE-1 Chapter 2 AE-46 Chapter 3 AE-76 Chapter 4 AE-115 Chapter 5 AE-151 Chapter 6 AE-196 Chapter 7 AE-250 Chapter 8 AE-295 Chapter 9 AE-292 Chapter 10 AE-367 Chapter 11 AE-399 Chapter 12 AE-421 Additional Exercises Answers AE-457 GROUP ACTIVITIES (per chapter) A-1 Chapter 1 A-1 Chapter 2 A-2 Chapter 3 A-3 Chapter 4 A-4 Chapter 5 A-5 Chapter 6 A-6 Chapter 7 A-7 Chapter 8 A-8 Chapter 9 A-9 Chapter 10 A-10 Chapter 11 A-11 Chapter 12 A-12 Group Activities Answers A-13

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    TEST FORMS C HAPTER 1 T ESTS (6 T ESTS ) T-1 Form A (FR) T-1 Form B (FR) T-3 Form C (FR) T-6 Form D (MC) T-9 Form E (MC) T-11 Form F (MC) T-14 C HAPTER 2 T ESTS (6 T ESTS ) T-17 Form A (FR) T-17 Form B (FR) T-21 Form C (FR) T-25 Form D (MC) T-29 Form E (MC) T-34 Form F (MC) T-39 C UMULATIVE R EVIEW 1-2 (2 T ESTS ) T-44 Form A (FR) T-44 Form B (MC) T-47 C HAPTER 3 T ESTS (6 T ESTS ) T-51 Form A (FR) T-51 Form B (FR) T-54 Form C (FR) T-57 Form D (MC) T-60 Form E (MC) T-64 Form F (MC) T-68 C HAPTER 4 T ESTS (6 T ESTS ) T-72 Form A (FR) T-72 Form B (FR) T-76 Form C (FR) T-80 Form D (MC) T-84 Form E (MC) T-89 Form F (MC) T-94 C UMULATIVE R EVIEW 1-4 (2 T ESTS ) T-99 Form A (FR) T-99 Form B (MC) T-102 C HAPTER 5 T ESTS (6 T ESTS ) T-107 Form A (FR) T-107 Form B (FR) T-110 Form C (FR) T-113 Form D (MC) T-116 Form E (MC) T-119 Form F (MC) T-122 C HAPTER 6 T ESTS (6 T ESTS ) T-125 Form A (FR) T-125 Form B (FR) T-127 Form C (FR) T-129 Form D (MC) T-131 Form E (MC) T-134 Form F (MC) T-137 C UMULATIVE R EVIEW 1-6 (2 T ESTS ) T-140 Form A (FR) T-140 Form B (MC) T-143

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    C HAPTER 7 T ESTS (6 T ESTS ) T-148 Form A (FR) T-148 Form B (FR) T-151 Form C (FR) T-154 Form D (MC) T-157 Form E (MC) T-161 Form F (MC) T-164 C HAPTER 8 TESTS (6 T ESTS ) T-167 Form A (FR) T-167 Form B (FR) T-171 Form C (FR) T-175 Form D (MC) T-178 Form E (MC) T-182 Form F (MC) T-186 C UMULATIVE R EVIEW 1-8 (2 T ESTS ) T-190 Form A (FR) T-190 Form B (MC) T-193 C HAPTER 9 TESTS (6 T ESTS ) T-197 Form A (FR) T-197 Form B (FR) T-200 Form C (FR) T-203 Form D (MC) T-206 Form E (MC) T-210 Form F (MC) T-214 C HAPTER 10 TESTS (6 T ESTS ) T-218 Form A (FR) T-218 Form B (FR) T-222 Form C (FR) T-226 Form D (MC) T-230 Form E (MC) T-235 Form F (MC) T-239 C UMULATIVE R EVIEW 1-10 (2 T ESTS ) T-242 Form A (FR) T-242 Form B (MC) T-245 C HAPTER 11 TESTS (6 T ESTS ) T-249 Form A (FR) T-249 Form B (FR) T-251 Form C (FR) T-253 Form D (MC) T-255 Form E (MC) T-257 Form F (MC) T-259 C HAPTER 12 TESTS (6 T ESTS ) T-273 Form A (FR) T-273 Form B (FR) T-275 Form C (FR) T-277 Form D (MC) T-279 Form E (MC) T-281 Form F (MC) T-283 F INAL (2 T ESTS ) T-286 Form A (FR) T-286 Form B (MC) T-292

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    TEST ANSWER KEYS T-299 Chapter 1 T-299 Chapter 2 T-301 Cumulative Review 1-2 T-305 Chapter 3 T-306 Chapter 4 T-309 Cumulative Review 1-4 T-313 Chapter 5 T-315 Chapter 6 T-317 Cumulative Review 1-6 T-318 Chapter 7 T-319 Chapter 8 T-321 Cumulative Review 1-8 T-324 Chapter 9 T-325 Chapter 10 T-327 Cumulative Review 1-10 T-331 Chapter 11 T-332 Chapter 12 T-336 Finals T-338

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    ML-1 Mini Lecture 1.1 Algebraic Expressions, Real Numbers and Interval Notation Learning Objectives: 1. Translate English phrases into algebraic expressions. 2. Evaluate algebraic expressions. 3. Use mathematical models. 4. Recognize the sets that make up the real numbers. 5. Use set-builder notation. 6. Use the symbols ∈ and ∉ . 7. Use inequality symbols. 8. Use interval notation. Examples: 1. Write each English phrase as an algebraic expression. Let x represent the number. a. Three less than five times a number. b. The product of a number and six, increased by four. 2. Evaluate each algebraic expression for the given value or values of the variable(s). a. 3 5 2 + + x x , for x = 2 b. ) ( 2 2 y x x + + , for x = 3 y = 4 3. Use the roster method to list the elements in each set. a. { x | x is an integer between 4 and 9} b. { x | x is an even whole number less than 10} 4. Use the meaning of the symbols ∈ and ∉ to determine whether each statement is true or false. a. 3 ∈ { x | x is a natural number} b. 9 ∉ {1, 3, 5, 7} 5. Write out the meaning of each inequality. Then determine whether the inequality is true or false. a. –10 > –8 b. 0 2 – ≤ c. 3 3 ≥ d. 5 – 2 ≤ 6. Express the interval [ ) 5, - ¥ in set builder notation and graph. Teaching Notes: • Be sure to go over important vocabulary for the section including: variable, algebraic expression, constant, exponential expression, equation, formula, natural numbers, whole numbers, integers, rational, irrational and real numbers. • Brainstorm the many words that translate to the four basic operations. Ex: increased - addition. • n b = b· b· … b (b appears as a factor “ n ” times). • Order of operation rules include: 1. First, perform all operations within grouping symbols. 2. Evaluate all exponential expressions. 3. Do all multiplication and division in the order in which they occur, working from left to right. 4. Last, do all additions and subtractions in the order in which they occur, working from left to right. • < is read “less than”, > is read “greater than” • ≤ is read “less than or equal to”, ≥ is read “greater than or equal to”

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    ML-2 Answers: 1. a. 3 – 5 x b. 4 6 + x 2. a. 17 b. 23 3. a. {5, 6, 7, 8} b. {0, 2, 4, 6, 8} 4. a. true b. true 5. a.–10 greater than –8, false b.–2 is less than or equal to 0, true c. 3 is greater than or equal to 3, true d. 2 is less than or equal to –5, false 6. { } | 5 x x ³ -

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    ML-3 Mini Lecture 1.2 Operations With Real Numbers and Simplifying Algebraic Expressions Learning Objectives: 1. Find a number’s absolute value. 2. Add real numbers. 3. Find opposites. 4. Subtract real numbers. 5. Multiply real numbers. 6. Evaluate exponential expressions. 7. Divide real numbers. 8. Use the order of operations. 9. Use commutative, associative, and distributive properties. 10. Simplify algebraic expressions. Examples: 1. Find the absolute value. a. 8 – b. 4 3 – c. 24 . 6 – d. 12 2. Add or subtract. a. –14 + 25 b. 3 1 – 4 3 – + c. 15 – (–10) d. 6.8 – 12.32 e. 8 3 – 8 5 – f. –52 + 52 g. –32 – (–38) h. 4.2 – (–8.1) 3. Evaluate. a. ( ) 2 8 – b. 2 8 – c. ( ) 4 3 – d. 4 3 – 4. Multiply or divide. a. 20 8 5 3 – ÷ b. (15) (–1) (–4) c. 0 24 – d. (–3.3) (1.2) e. ( )( )( )( ) 2 0 7 6 − f. 18 0 g. 3 2 8 − ÷ − h. 15 14 7 3 − ⋅ 5. Use the distributive property and simplify. a. 6( x –2) b. –3 (6 – y ) c. –4( x – 5 – y ) 6. Rewrite to show how the associative property could be used to simplify the expression. Then simplify. a. 6(–4 x ) b. ( x + 124) + 376 7. Simplify using the order of operation. a. 6 4 20 5 + ÷ ⋅ b. 2 2 – 15 ) 2 ( 5 – ) 2 (– 6 c. ( ) 6 3 2 3 x x - -

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    ML-4 Teaching Notes: • Remind students that absolute value measures distance from zero, and for that reason, it is always positive. • Opposites and additives inverses are just different names for the same thing. • Students need to be reminded often that a negative sign is only part of the base if it is inside parentheses with the base. • When opposites are added, the result is zero. • Make sure students understand what is behind subtraction – why subtraction can be changed to addition of the opposite. • Never, never, never multiply the base and the exponent together! Students are often tempted to do this. Answers: 1. a. 8 b. 4 3 c. 6.24 d. 12 2. a. 11 b. 12 1 1 – 12 13 – or c. 25 d. –5.52 e. –1 f. 0 g. 6 h. 12.3 3. a. 64 b. –64 c. 81 d. –81 4. a. 2 3 – b. 60 c. undefined d. –3.96 e. 0 f. 0 g. 12 h. 5 2 − 5. a. 6 x – 12 b. –18 + 3 y c. –4 x + 20 + 4 y 6. a. x x 24 – ) 4 – 6 ( = ⋅ b. 500 ) 376 124 ( + = + + x x 7. a. 31 b. –2 c. 15 x – 12

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    ML-5 Mini Lecture 1.3 Graphing Equations Learning Objectives: 1. Plot points in the rectangular coordinate system. 2. Graph equations in the rectangular coordinate system. 3. Use the rectangular system to visualize relationships between variables. 4. Interpret information about a graphing utility’s viewing rectangle or table. Examples: 1. Plot the following point in a rectangular coordinate system. A. (–2, 3) B. (–4, 0) C. (1, 5) D. (–1, –4) E. (3, –3) F. (0, 2) 2. Complete the table of values for 3 – x y = , then graph the equation. 3. Complete the table of values for 2 – 2 x y = , then graph the equation. 4. Complete the table of values for , 1 – x y = then graph the equation. x y = | x –1| ( x, y ) –4 –3 –2 –1 0 1 2 3 4 x y = x – 3 ( x, y ) –2 –1 0 1 2 x y = 2 – x 2 ( x, y ) –3 –2 –1 0 1 2 3

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    ML-6 Teaching Notes: • The point plotting method is one method for graphing equations. • The graph of a linear equation is a line. • The graph of a quadratic equation is a parabola. • The graph of an absolute value equation is a “V” shape that can shoot upward or downward. Answers: 1. 2. (–2, –5) (–1, –4) (0, –3) (1, –2) (2, –1) 3. (–3, –7) (–2, –2) (–1, 1) (0, 2) (1, 1) (2, –2) (3, –7) 4. (–4, 5) (–3, 4) (–2, 3) (–1, 2) (0, 1) (1, 0) (2, 1) (3, 2) (4, 3) Figure for Answer 3 Figure for Answer 4

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    ML-7 Mini Lecture 1.4 Solving Linear Equations Learning Objectives: 1. Solve linear equations. 2. Recognize identities, conditional equations, and inconsistent equations. 3. Solve applied problems using mathematical models. Examples: Solve each equation. If fractions are involved, you may want to clear the fractions first. 1. a. 22 7 5 = + x b. x x 7 3 32 = + c. 9 – 2 3 = + x 2. a. ) 1 – 2 ( 4 ) 2 3 ( 5 x x = + b. ) 4 – ( 5 ) 6 – 2 ( – ) 4 – ( 5 x x x = c. ) 5 – ( 3 – 25 x x = 3. a. 10 7 – 5 4 x x = b. 6 5 3 – 2 – 8 1 = + a a c. 1 5 1 – 3 – 3 4 – = y y d. ) 5 – 3 ( 5 1 ) 8 2 ( 3 1 x x = + Solve and determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4. a. x x x 3 ) 4 – ( 2 3 5 + = + b. ) 4 ( 4 8 8 ) 4 – 2 ( 6 + + = + x x x c. 3 ) 1 ( 5 ) 4 ( 2 3 + + = + + a a a d. ) 3 3 ( 8 ) 4 4 ( 6 + = + y y e. 14 – 4 . 1 10 – 6 . 0 x x = f. 0 ) – 2 ( 3 ) 1 – ( 6 = + x x Teaching Notes: • Students may need to be reminded there is no “right” or “wrong” side of the equation. Some students have a problem when the variable ends up on the right side of the equation. • Students need to practice clearing equations of fractions by multiplying each term (whether it is a fraction or not) by the least common denominator of all the terms. Answers: 1. a. 3 b. 8 c. –6 2. a. –2 b. 3 c. 10 3. a. 10 b. 3 c. –8 d. –55 4. a. Inconsistent ; No Solution b. Inconsistent; No Solution c. Identity; infinitely many solutions d. Identity; infinitely many solutions e. 5; conditional equation f. 0; conditional equation

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    ML-8 Mini Lecture 1.5 Problem Solving and Using Formulas Learning Objectives: 1. Solve algebraic word problems using linear equations. 2. Solve a formula for a variable. Examples: Solve the following using the five step strategy for solving word problems. 1. When 12 is subtracted from three times a number, the result is 36. What is the number? 2. 15% of what number is 255? 3. In a triangle, the measure of the third angle is twice the measure of the first angle. The measure of the second angle is twenty more than the first. Find the measure of each angle. 4. The dog run is six feet longer than it is wide and the perimeter measures 32 feet. Determine the measurements of the length and width of the dog run. 5. A new automobile sells for $28,000. If the mark-up is 25% of the dealer’s cost, what is the dealer’s cost? Solve each formula for the specified variable. 6. Bh V 3 1 = for B 7. ) ( rt l P A + = for P 8. ) 2 – ( 180 n S = for n 9. 2 d KMm f = for M 10. C By Ax = + for A Teaching Notes: • Use the five step strategy for solving word problems. • Read the problem carefully. Let a variable represent one of the quantities in the problem. • If necessary, write an expression for any other unknown quantities in the problem in terms of the same variable used in step 1. • Write an equation to describe the conditions of the problem. • Solve the equation and answer the problem’s question. • Remind students to always check to make sure their answer makes sense. Answers: 1. 16 ; 36 12 – 3 = = x x 2. 1700 ; 255 15 . 0 = = x x 3.    80 , 60 , 40 ; 180 2 ) 20 ( = + + + x x x 4. long feet 11 6 x , feet wide 5 ; 32 ) 6 ( 2 2 = + = = + + x x x 5. 400 , 22 $ ; 000 , 28 25 . 0 = = + x x x 6. h V B 3 = 7. rt l A P + = 8. 2 180 + = S n 9. Km fd M 2 = 10. x By C A – =

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    ML-9 Mini Lecture 1.6 Properties of Integral Exponents Learning Objectives: 1. Use the product rule. 2. Use the quotient rule. 3. Use the zero-exponent rule. 4. Use the negative exponent rule. 5. Use the power rule. 6. Find the power of a product. 7. Find the power of a quotient. 8. Simplify exponential expressions. Examples: Simplify. Final answers should not contain any negative exponents. 1. a. 3 4 y y ⋅ b. ) 3 )( 4 ( 4 a a c. ) 4 )(– ( 3 4 xyz z xy d. ) 6 )( ( 3 – 3 5 3 2 1 n m n m 2. a. 4 10 y y b. 2 6 5 25 a a c. 6 – 3 x x d. 2 – 4 2 6 4 6 18 z y x z y x 3. a. 2 – 4 b. 3 – 5 1 c. 0 6 d. 0 6 x e. 3 – 2 – 3 y x f. 5 – 4 7 y x g. 8 – 2 – a a h. 1 – 15 4. a. 2 10 ) ( x b. 3 – 6 – ) ( y c. 1 – 2 ) 4 ( d. 2 4 ) ( a 5. a. 2 0 2 ) 3 ( b a b. 2 – 2 – 3 – ) 5 ( y x c. 2 – 3 2       d. 2 3 – 2 2 6         y x 6. a. 2 2 3 – 6 ) 4 )( 5 ( y x y x b. 2 – 4 4 – 2 – 2 – 3 ) 6 ( ) 2 ( y x y x c. 2 – 3 – 3 4 ) 2 ( 5 y x y x d. 3 – 6 4 1       b a Teaching Notes: • Exponent rules are very easy as presented – one at a time. Students often become confused when several rules are used in one problem. Constant reinforcement and lots of practice will help. • Remind students that when a variable appears to have no exponent – there is an invisible exponent of one. • Never, never, never multiply a base and an exponent together. • Always (exception: scientific notation) write final answers with positive exponents only. Answers: 1. a. 7 y b. 5 12 a c. 4 5 2 4 – z y x d. 2 6 3 n m 2. a. 6 y b. 4 5 a c. 9 x d. 3 2 2 3 z y x 3. a. 16 1 b. 125 c. 1 d. 6 e. 3 2 3 y x f. 5 4 7 y x g. 6 a h. 15 1 4. a. 20 x b. 18 y c. 16 1 d. 8 a 5. a. 4 9 a b. 25 4 6 y x c. 4 9 d. 6 4 9 y x 6. a. y x 10 80 b. 14 12 9 x y c. 2 5 20 x y d. 18 12 b a

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    ML-10 Mini Lecture 1.7 Scientific Notation Learning Objectives: 1. Convert from scientific to decimal notation. 2. Convert from decimal to scientific notation. 3. Perform computations with scientific notation. 4. Use scientific notation to solve problems. Examples: 1. Write each number in decimal notation. a. 5 10 4 . 3 – × b. 4 – 10 015 . 2 × 2. Write each number in scientific notation. a. 32,500,000,000 b. –0.00417 c. 4 10 9432 × 3. Perform the indicated computations, writing the answers in scientific notation. a. ) 10 8 )( 10 4 . 2 ( 5 – 3 × × b. 2 – 4 10 4 10 8 . 6 × × 4. In Central City, the population is 176,000. Express the population in scientific notation. Teaching Notes: • A number is written in scientific notation when it is expressed in the form n a 10 × with 1 ≤ | a | < 10 and “ n ” is an integer. • When multiplying terms written in scientific notation m n m n b a b a + × × = × × 10 ) ( ) 10 )( 10 ( . • When dividing terms written in scientific notation . 10 10 10 – n m n m b a b a × = × × • When multiplying or dividing is complete, make sure the final answer is in scientific notation. • Students need to be reminded that a number must be written as a number between 1 and 10 to be in scientific notation. • The sign of a number has nothing to do with the sign of the power when a number is written in scientific notation. Answers: 1. a. –340,000 b. 0.0002015 2. a. 10 10 25 . 3 × b. 3 – 10 17 . 4 × c. 7 10 432 . 9 × 3. a. 1 – 10 92 . 1 × b. 6 10 7 . 1 × 4. 5 10 76 . 1 ×

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    ML-11 Mini Lecture 2.1 Introduction to Functions Learning Objectives: 1. Find the domain and range of a relation. 2. Determine whether a relation is a function. 3. Evaluate a function. Examples: 1. Find the domain and range of the relation. a. {(1, 5), (2, 10), (3, 15), (4, 20), (5, 25)} b. {(1, -1), (0, 0), (-5, 5)} 2. Determine whether each relation is a function. a. { } ) 10 , 5 ( ), 9 , 5 ( ), 8 , 5 ( ), 7 , 5 ( ), 6 , 5 ( b. {(5, 6), (6, 7), (7, 8), (8, 9), (9, 10)} 3. Find the indicated function value. a. 2 – 3 ) ( for ) 3 ( x x f f = b. 4 – 2 ) ( for ) 2 (– 2 + = x x x g g c. 2 3 – ) ( for ) 1 (– 2 + = t t t h h d. 3 2 ) ( for ) ( + = + x x f h a f 4. Function g is defined by the table Find the indicated function value. a. g (2) b. g (4) Teaching Notes: • A relation is any set of ordered pairs. • The set of all first terms “ x -values” of the ordered pairs is called the domain . • The set of all second terms “ y -values” of the ordered pairs is called the range . • A function is a relation in which each member of the domain corresponds to exactly one member of the range. • A function is a relation in which no two ordered pairs have the same first component and different second components. • The variable “ x ” is called the independent variable because it can be assigned any value from the domain. • The variable “ y ” is called the dependent variable because its value depends on “ x ”. • The notation f(x) , read “ f of x ” represents the value of the function at the number “ x ”. Answers: 1. domain {1, 0, -5} range {-1, 0, 5} 2. a. not a function b. function 3. a. 7 b. 14 c. 6 d. 3 2 2 + + h a n 4. a. 6 b. 10 x g (x) 0 2 1 4 2 6 3 8 4 10

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    ML-12 X Y -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 0 Mini Lecture 2.2 Graphs of Functions Learning Objectives: 1. Graph functions by plotting points. 2. Use the vertical line test to identify functions. 3. Obtain information about a function from its graph. 4. Identify the domain and range of a function from its graph. Examples: State the domain of each function. 1. Graph the function 1 3 ) ( and 3 ) ( + = = x x g x x f in the same rectangular coordinate system. Graph integers for x starting with –2 and ending with 2. How is the graph of g related to the graph of f ? 2. Use the vertical line test to identify graphs in which y is a function of x . a. b. c. 3. Use the graph of f to find the indicated function value. a. f (2) b. f (0) c. f (1) 4. Use the graph each function to identify its domain and range. a. b. X Y -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 0 (-5 ,1) (1, 1) (-2, 1) (4, 1) X Y -6 -4 -2 2 4 6 -6 -4 -2 2 4 6 0

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    ML-13 Teaching notes: • The graph of a function is the graph of the ordered pairs. • If a vertical line intersects a graph in more than one point, the graph does not define y as a function of x . Answers : 1. The graph of g is the graph of f shifted up 1 unit. 2. a. yes b. no c. yes 3. a. 0 b. 4 c. 1 4. a. Domain: { } 5,1, 1.4 - Range: { } 1 b. Domain: [ ) 0, ¥ Range: [ ) 2, ¥

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    ML-14 Mini Lecture 2.3 The Algebra of Functions Learning Objectives: 1. Find the domain of a function. 2. Use the algebra of functions to combine functions and determine domains. Examples: State the domain of each function. 1. a. ( ) 3 1 f x x = - b. ( ) 4 2 x g x x = - c. ( ) 2 6 h x x x = + - d. ( ) 1 7 5 9 p x x x = + + - 2. Let ( ) x x x f 2 2 − = and ( ) 3 + = x x g . Find the following; a. ( )( ) x g f + b. the domain of g f + c. ( )( ) 2 − + g f 3. Let ( ) 2 5 + = x x f and ( ) 6 1 g x x = - . Find the following; a. ( )( ) x g f + b. The domain of g f + 4. Let ( ) 1 2 + = x x f and ( ) 3 = + x x g . Find the following; a. ( )( ) x g f + b. ( )( ) 2 − + g f c. ( )( ) x g f − d. ( )( ) 0 g f − e. ( ) 2 −         g f Teaching Notes: • Students need to be reminded that division by zero is undefined. The value of “ x ” cannot be anything that would make the denominator of a fraction zero. • Students often exclude values from the domain that would make the numerator zero, warn against this. • Show students why the radicand of a square root function must be greater than or equal to zero. This is a good place to use the graphing calculator so students can “see” what happens. Answers: 1. a. ( ) , -¥ ¥ b. ( ) ( ) , 2 or 2, -¥ ¥ c. ( ) ( ) , 6 or 6, -¥ ¥ d. ( ) ( ) ( ) , 5 or 5, 9 or 9, -¥ - - ¥ 2. a. 2 3 x x - + b. ( ) , -¥ ¥ c. 3 4. a. 5 6 2 1 x x + + - b. ( ) ( ) ( ) , 2 or 2, 1 or 1, -¥ - - ¥ 4. a. 2 2 − + x x b. 0 c. 4 2 + − x x d. 4 e. -1

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    ML-15 Mini Lecture 2.4 Linear Functions and Slope Learning Objectives: 1. Use intercepts to graph a linear function in standard form. 2. Compute a line’s slope. 3. Find a line’s slope and y -intercept from its equation. 4. Graph linear functions in slope-intercept form. 5. Graph horizontal or vertical lines. 6. Interpret slope as rate of change. 7. Find a function’s average rate of change. 8. Use slope and y -intercept to model data. Examples: 1. Use intercepts and a checkpoint to graph each linear function. Name the x -intercept and the y -intercept. a. 10 5 2 = + y x b. – 2 4 x y = 2. Find the slope of the line passing through each pair of points. Then indicate whether the line through the points rises, falls, is horizontal, or is vertical. a. (2, 5) and (–6, 3) b. ) 3 , 1 ( and ) 0 , 5 ( c. (3, 0) and (3, 4) - d. (2, 4) and ( 6, 4) - 3. a. Find the slope and y -intercept for the line whose equation is 12 4 3 = + y x and then graph the equation. b. Find the slope and y -intercept for the linear function 3 2 1 ) ( + = x x f and then graph the function. 4. Graph the linear equations. a. 2 = x b. 12 – 3 = y Teaching Notes: • The standard form of the equation of a line is C By Ax = + , as long as A and B are not both zero. • A x -intercept will have a corresponding y coordinate of 0. • A y -intercept will have a corresponding x coordinate of 0. • The slope of a line compares the vertical change to the horizontal change ( ) run rise . • Slope formula is: 1 2 1 2 – – x x y y m = . • A line that rises from left to right has a positive slope. • A line that falls from left to right has a negative slope. • A line that is horizontal has zero slope. • A line that is vertical has an undefined slope. • The slope-intercept form of the equation of a line is b mx y + = where m is the slope and b is the y -intercept.

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    ML-16 Answers: 1. a. b. 2. a. 4 1 = m , rises b. 3 4 m = - , falls c. undefined, vertical d. 0, horizontal 3. a. 3) (0, intercept - 4 3 – 3 4 3 – y m x y = + = b. 1 2 m = - y -intercept (0,3) 4. a. b.

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    ML-17 Mini Lecture 2.5 The Point-Slope Form of the Equation of a Line Learning Objectives: 1. Use the point-slope form to write equations of a line. 2. Model data with linear functions and make predictions. 3. Find slopes and equations of parallel and perpendicular lines. Examples: 1. Write an equation in point-slope form for the line with the given information. a. slope = –3, passing through (–4, 1) b. slope = 3 2 , passing through (6, 3) c. slope = 0, passing through (–3, 2) 2. Write an equation in point slope form for the line with the given information. Then write the equation in slope-intercept form. a. slope = 4 1 , passing through (4, 3) b. passing through (1, 5) and (3, –5) c. passing through (–2, 4) and (2, 6) 3. Find the slope of a line parallel to each given line. a. 7 4 3 + = x y b. 6 3 – 2 = y x c. 8 2 = + y x 4. Find the slope of a line perpendicular to each given line. a. 1 4 – + = x y b. 12 3 = + y x c. 10 2 – 4 = y x Write an equation for each of the following in slope-intercept form. 5. A line passing through the origin (0, 0) and parallel to a line whose equation is 5 – 4 x y = . 6. A line passing through (4, –4) and parallel to a line whose equation is 6 2 = + y x . 7. A line passing through (6, –1) and perpendicular to a line whose equation is 1 3 – = y x . 8. A line passing through (2, –5) and perpendicular to a line whose equation is 0 7 2 – 4 = + x y . Teaching Notes: • The point-slope form of the equation of a non-vertical line with slope m that passes through the point ( ) y x , is ( ) 1 1 x x m y y − = − . • Make sure students memorize the point-slope form of a linear equation and know what each letter represents. • Students will need lots of practice on this. • This is a good time for students to be able to visualize parallel line lines and “see” that they have the same slopes, but different y -intercepts.

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    ML-18 Answers: 1. a. ) 4 ( 3 – 1 – + = x y b. 6 – ( 3 2 3 – x y = ) c. ) 3 ( 0 2 – + = x y 2. a. ) 4 – ( 4 1 3 – x y = ; 2 4 1 + = x y b. ) 1 – ( 5 – 5 – x y = or ) 3 – ( 5 – 5 x y = + ; 10 5 – + = x y c. ) 2 – ( 2 1 6 – x y = or ) 2 ( 2 1 4 – + = x y ; 5 2 1 + = x y 3. a. 4 3 b. 3 2 c. 2 1 – 4. a. 4 1 b. 3 1 c. 2 1 – 5. x y 4 = 6. 2 – 2 1 – x y = 7. 17 3 – + = x y 8. 1 – 2 – x y =

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    ML-19 Mini Lecture 3.1 Systems of Linear Equations in Two Variables Learning Objectives: 1. Determine whether an ordered pair is a solution of a system of linear equations. 2. Solve systems of linear equations by graphing. 3. Solve systems of linear equations by substitution. 4. Solve systems of linear equations by addition. 5. Select the most efficient method for solving a system of linear equations. 6. Identify systems that do not have exactly one ordered-pair solution. Examples: 1. Determine whether (2, –3) is a solution of the system. 2 4 2 1 x y x y ì + = ï ï í ï + = - ï î 2. Solve by graphing: –3 0 2 x y y x ì + = ï ï í ï = + ï î 3. Solve by the substitution method. a. 2 9 3 4 9 y x x y ì = + ï ï í ï - = ï î b. 3 5 12 4 11 x y x y ì + = ï ï í ï + = ï î 4. Solve by the addition method. a. 2 –9 3 5 4 x y x y ì + = ï ï í ï + = ï î b. 5 3 4 2 1 2 6 6 x y x y + = = = - 5. Solve by the method of your choice. Identify inconsistent systems and systems with dependent equations. a. – 2 1 3 6 2 x y x y ì = ï ï í ï - = ï î b. 3 – –3 6 2 6 x y x y ì = ï ï í ï- = = ï î Teaching Notes: • A system of linear equations in two variables represents a pair of lines. There are three possibilities for solutions: a. If two lines intersect at one point, then there is exactly one ordered-pair solution. b. If two lines are parallel, then there is no solution. c. If two lines are identical, then there are infinitely many solutions. • All three methods for solving systems of linear equations in two variables will produce the same answer; however, one method will sometimes be more efficient than another. Answers: 1. no 2. (1, 3) 3. a. (–9, –9) b. (–1, 3) 4. a. (–7, 5) b. (–1, 2) 5. a. Inconsistent; the lines are parallel, no solution b. {( x , y ) | 3 x – y = –3} or {( x , y ) | –6 x + 2 y = 6} the lines coincide and the system has infinitely many solutions.

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    ML-20 Mini Lecture 3.2 Problem Solving and Business Applications Using Systems of Equations Learning Objectives: 1. Solve problems using systems of equations. 2. Use functions to model revenue, cost, and profit, and perform a break even analysis. Examples: 1. At JoJo’s pet and supply store female gerbils sell for $8 each and male gerbils sell for $5 each. On Saturday 20 gerbils were sold for a total of $139. How many males and how many females were sold on Saturday? 2. To promote the grand opening of Leroy’s Toys the store gave away 9000 miniature cars and goofy sunglasses. The cars cost 13¢ and the sunglasses cost 15¢ each. Leroy spent a total of $1290 on the giveaways. How many of each item did he buy? 3. The larger of two numbers is equal to three times the smaller. If twice the larger is added to three times the smaller, the sum is 27. Find the numbers. 4. Twice the length of a rectangle is equal to five times its width. The perimeter of the rectangle is 77 inches. Find the dimensions of the rectangle. 5. Cashews cost $3.60 per pound and almonds cost $2.70 per pound. For a fundraiser, the volleyball team will be selling bags of mixed nuts. How many pounds of cashews and how many pounds of almonds should the team buy in order to make a 60 pound mixture that will sell for $3.00 per pound? 6. How many gallons of 15% alcohol solution and how many gallons of 40% alcohol solution should be mixed to get 20 gallons of a 30% alcohol solution? 7. Joe and Jack inherited $200,000 from their Aunt Lulu. They each decided to put their money into savings accounts for 1 year and then decide how to spend it. Joe’s money earned 5% interest and Jack’s earned 3.8%. Together, their money earned $8,608 in interest. How much did each boy inherit? 8. A small airplane can travel 600 miles in 4 hours with the wind. The return trip against the wind takes 5 hours. Find the speed of the plan in still air and the speed of the wind. 9. Since Jane’s grandparents enjoy making birdhouses and selling them at local craft shows. They will pay $150 for the booth rental for the weekend. The materials for making each birdhouse cost $8.75. If they are able to sell the birdhouses for $15.00 each, how many will they need to sell to beak even? What if they sell 15 birdhouses? 40 birdhouses? Answers: 1. 13 females; 7 males 2. 6000 sunglasses; 3000 cars 3. 3 and 9 4. 11 inches wide; 27.5 inches long 5. 20 lbs. cashews’ 40 lbs. almonds 6. 12 gallons of 40%; 8 gallons of 15% 7. Joe $84,000; Jack $116,000 8. 135 mph; 15 mph wind 9. 24 birdhouses to break even; they will lose $56.25; they will make $100.

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    ML-21 Mini Lecture 3.3 Systems of Linear Equations in Three Variables Learning Objectives: 1. Verify the solution of a system of linear equations in three variables. 2. Solve systems of linear equations in three variables. 3. Identify inconsistent and dependent systems. 4. Solve problems using systems in three variables. Examples 1. Show that the ordered triple (1, 2, 3) is a solution of the system: 6 2 3 2 3 4 x y z x y z x y z ì + + = ï ï ï ï - + = í ï ï + - = - ï ï î 2. Solve the system: a. 4 2 2 2 2 x y z x y z x y z ì + + = ï ï ï ï - - = í ï ï + - = ï ï î b. – 4 – 3 2 2 8 6 1 3 0 x y z x y z x y z ì + = ï ï ï ï - + = í ï ï - + = ï ï î c. 2 3 – 5 4 6 2 10 4 3 5 x y z x y z x y z ì + = ï ï ï ï + - = í ï ï - + = ï ï î 3. Create three equations from the stated problem and then solve. The sum of the three numbers is 14. The largest is 4 times the smallest, while the sum of the smallest and twice the largest is 18. 4. Find the quadratic function 2 y ax bx c = = = whose graph passes through the points (1, 3), (2, 5), and ( – 1, 11). Teaching Notes: • A system of linear equations is three variables represents three planes. • A linear system that intersects at one point is called a consistent system and has an ordered triple as an answer ( x, y, z). • A linear system that intersects at infinitely many points is also called a consistent system and is also called dependent . • A linear system that has no common point(s) of intersection represents an inconsistent system and has no solution. Answers: 1. 1+2+3=6 , 2–2+3=3 , 1+4–9= –4 2. a. (3,–1,2) b. No Solution , Inconsistent system. c. infinitely many solutions, dependant equations 3. 14 = + + z y x , x z 4 = , 18 2 = + z x . The numbers are 2, 4 and 8. 4. 2 2 4 5 y x x = - + or ( ) 2 2 4 5 f x x x = - +

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    ML-22 Mini Lecture 3.4 Matrix Solutions to Linear Systems Learning Objectives: 1. Write the augmented matrix for a linear system. 2. Perform matrix row operations. 3. Use matrices to solve linear systems in two variables. 4. Use matrices to solve linear systems in three variables. 5. Use matrices to identify inconsistent and dependent systems. Examples: 1. Write the matrix for each system. a. 3 – 4 12 6 5 x y x y ì = ï ï í ï + = ï î b. 2 7 4 8 x y x ì + = ï ï í ï = ï î c. 2 3 0 3 4 3 2 5 10 x y z x y z x y z ì + + = ï ï ï ï - + = í ï ï + - = ï ï î 2. Write the system of linear equations represented by each augmented matrix. a.       11 1 1 2 1 – 1 b.       0 6 1 1 3 – 2 c.           1 – 11 – 3 2 2 1 2 – 1 1 – 1 1 4 3. Perform each matrix row operation as indicated and write the new matrix. a.       4 – 13 2 1 3 – 2 b.       13 4 – 3 – 2 2 1 c.           8 – 2 5 2 – 3 1 1 2 1 2 1 – 2 – 2 1 R R ↔ 2 1 2 – R R + 2 1 2 1 2 and R R R R + ↔ 4. Solve each system using matrices. a. 3 10 1 2 1 x y x y ì + = ï ï í ï + = - ï î b. 3 5 3 3 5 2 2 9 x y x x y z x y z ì + + = ï ï ï ï + - = í ï ï + - = ï ï î c. 2 2 1 2 3 2 4 0 x y z x y z x y z ì + + = ï ï ï ï - + + = í ï ï + + = ï ï î Teaching Notes: • Organization and neatness is very important when using matrices to solve systems. • Caution students to watch signs carefully. • Using the calculator is a great way to check systems solved using matrices. • Students tend to panic if fractions “happen”. Encourage them to keep working through the problem.

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    ML-23 Answers: 1. a.       5 12 6 1 4 – 3 b.       8 7 0 4 1 2 c.           10 3 0 5 – 1 2 1 4 – 3 3 2 1 2. a. 11 2 1 – = + = y x y x b. 0 6 3 – 2 = + = y x y x c. 1 – 2 2 11 – 2 – – 3 4 = + + = + = + + z y x z y x z y x 3. a.       13 4 – 3 – 2 2 1 b.       21 4 – 7 – 0 2 1 c.           8 – 9 2 2 – 3 1 4 3 0 1 2 1 4. a. (–3, 1) b. (1, 2, –2) c. (–2, 3, –1)

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    ML-24 Mini Lecture 3.5 Determinants and Cramer’s Rule Learning Objectives: 1. Evaluate a second-order determinant. 2. Solve a system of linear equations in two variables using Cramer’s rule. 3. Evaluate a third-order determinant. 4. Solve a system of linear equations in three variables using Cramer’s rule. 5. Use determinants to identify inconsistent and dependent systems. Examples: 1. Evaluate the determinant of each of the following matrices: a.       3 2 0 1 b.       5 2 – 3 – 8 2. Use Cramer’s rule to solve the system: 2 – 3 3 4 2 10 x y x y ì = ï ï í ï - = ï î 3. Evaluate the determinant of the following matrix:           2 1 0 2 – 0 1 0 1 – 2 4. Use Cramer’s rule to solve the system: – 3 6 2 7 2 3 4 x y z x y z x y z ì + + = ï ï ï ï + + = í ï ï + + = ï ï î 5. Use Cramer’s rule to solve each system or to determine that the system is inconsistent or contains dependent equations. a. 2 – 4 5 2 3 x y x y ì = ï ï í ï- + = ï î b. 2 6 16 3 8 x y x y ì + = ï ï í ï + = ï î Teaching Notes: • A matrix of order n m × has m rows and n columns. • The determinant of a 2 x 2 matrix       2 2 1 1 b a b a is denoted by       2 2 1 1 b a b a = 1 2 2 1 – b a b a . • If the determinant D 1 = 0 and at least one of the determinants in the numerator is not 0, then the system is inconsistent and there is no solution . • If the determinant, D 1 = 0 and all the determinants in the numerators are 0, then the equations in the systems are dependent and the system has infinitely many solutions. Answers: 1. a. 3 b. 34 2. (3, 1) 3. 6 4. (2, –1, 3) 5. a. inconsistent b. dependent equations; infinitely many solutions

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    ML-25 Mini Lecture 4.1 Solving Linear Inequalities Learning Objectives: 1. Solve linear inequalities. 2. Recognize inequalities with no solution or all real numbers as solutions. 3. Solve applied problems using linear inequalities. Examples: 1. Solve and graph the solution set on a number line. Give solutions in interval notation. a. 2 x –5 ≥3 b. 3 x – 5 ≤ 6 x + 4 c. 4 1 + x > 4 1 – 2 x + 8 3 d. 4( x + 1) > 4 x + 2 e. 2 x + 2 ≤ 2 x – 2 Teaching Notes: • One method of representing the solution set of an inequality is with interval notation . Using the notation x ≥ –3 is expressed as [–3, ∞). The bracket indicates –3 is included in the interval. The infinity symbol does not represent a real number and the interval extends indefinitely to the right. • When multiplying or dividing both sides of an inequality by a negative quantity, remember to reverse the direction of the inequality symbol. • When an inequality has been solved and the variable has been eliminated and the result is a false statement, the inequality has no solution, Ø. • When an inequality has been solved and the variable has been eliminated and the result is a true statement, the solution for the inequality is all real numbers. Answers: 1. a. [4, ∞) b. [–3, ∞) c. (–∞, 2 1 ) d. (–∞, ∞) e. Ø

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