Lecture Guide. Math 90  Intermediate Algebra. Stephen Toner. Intermediate Algebra, 2nd edition. Miller, O'Neill, & Hyde. Victor Valley College


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1 Lecture Guide Math 90  Intermediate Algebra to accompan Intermediate Algebra, 2nd edition Miller, O'Neill, & Hde Prepared b Stephen Toner Victor Valle College Last updated: 11/24/10 0
2 1.1 Sets of Numbers The set of rational numbers can be written as A. Tpes of Numbers Natural  Whole  Integers  Rational  an number which can be epressed as the of two numbers, provided the denominator is not zero. Aleks Problem: Classif each number below as an integer or not. Integer? Yes No Aleks Problem: Classif each number below as irrational or rational Rational Irrational Irrational  Real  B. Set Builder Notation can be written as Aleks Problem: Rewrite the Set R b listing the elements. Make sure to use the appropriate set notation. 1
3 C. Interval Notation D. Unions and Intersections Inequalit Graph Interval Notation Given:, Find each of the following: Find each union or intersection. Write our solution using interval notation
4 5. E. Words and Inequalities more than less than at least at most 6. no more than no less than 7. Aleks Problem: The sets C and D are defined as follows: Write and using interval notation. 8. Aleks Problem: Write " is less than or equal to b, and 3 is greater than b" as an algebraic epression. 3
5 1.2 Operations with Real Numbers 1.2 #84 Order of Operations We assume ou remember the rules for adding, subtracting, multipling and dividing real numbers... Evaluate: (57 of them) 1.2 #92 Order of Operations Propert: Be careful! *Evaluate each if and : a. b. 4
6 1.3 Simplifing Epressions A. Properties Aleks: A trick problem... Which propert is illustrated? Commutative Propert of addition... of multiplication Simplif: Associative Propert of addition... of multiplication 1.3 #68 Simplif: Additive Identit Multiplicative Identit 1.3 #74 Additive Inverse Multiplicative Inverse Distributive Properties 5
7 1.4 Linear Equations 1.4 #56 Solve: Solve: 1.4 #42 Solve: There are three classes of equtions: Conditional the tpe we've been doing Identit "all real numbers" is our solution Contradiction there is no solution 1.4 #72 Solve: When an equation contains fractions, it is usuall a good idea to multipl through b the least common denominator to clear the fractions. 1.4 #46 Solve: 1.4 #80 Solve: 6
8 1.5 Word Problems A. Number Problems 1.5 #16 Twice the sum of a number and three is the same as 1 subtracted from the number. Find the number. 1.5 #26 Five times the smallest of three consecutive even integers is 10 more than twice the largest. Find the integers. C. Percentage Word Problems B. Consecutive Integer Problems consecutive integers 1.5 #26 The price of a used tetbook after a 35% markdown is $ What was the original price? consecutive odd integers consecutive even integers 1.5 #24 Four times the smaller of two consecutive odd integers is the same as 73 less than 5 times the larger. Find the integers. 1.5 #32 Robert can take out a 3r loan at 8% simple interest or a 2r loan at 8.5% simple interest. If he borrows $7000, how much interest will he pa for each loan? Which option will require less interest? 7
9 D. Miture Problems 1.5 #58 A nut miute consists of almonds and cashews. Almonds are $4.98/lb. and cashews are $6.98/lb. How man pounds of each tpe of nut must be mied to produce 16 pounds of a miture selling for $5.73/lb.? 1.5 #52 One fruit punch has 40% fruit juice and another is 70% fruit juice. How much of the 40% punch should be mied with 10 gallons of 70% punch to create a fruit punch that is 45% fruit juice? 1.5 #56 How man ounces of water must be added to 20 oz. of an 8% salt solution to make a 2% salt solution? 8
10 Aleks problem: Two factor plants are making TV panels. Plant A produced 3000 fewer panels than Plant B did. Five percent of the panels from Plant A and 2% of the panels from Plant B are defective. How man panels did Plant A produce, if the two plants together produced 1110 defective panels? E. Investment Problems 1.5 #44 Amanda borrowed $6000 from two sources: her parents and a credit union. Her parents charged her 3% simple interest and the credit union charged 8%. If after 1 ear she paid $255 in interest, how much did she borrow from each source? F. Distance Problems Aleks problem: A car travels 221 miles in 3 hours and 15 minutes. How man miles did it travel per hour? 9
11 Distance Problems General Procedure: 1.5 #62 Two cars are 190 miles apart and travel toward each other. The meet in 2 hours. One car travels 5 miles per hour slower than the other car. What are the speeds of both cars? 1.5 #60 A woman can hike 1 mph faster down a trail than she can on the return trip uphill. It takes 3 hours to get to her destination and 6 hours to return. What is her downhill speed? 1.5 #64 Two canoes travel down a river starting at 9:00. One canoe travels twice as fast as the other. After 3.5 hours the canoes are 5.25 miles apart. Find the average rate of each canoe. 10
12 1.6 Literal Equations & Geometr Problems A. Literal Equations 1.6 #36 Solve for. B. Geometr Problems 1.6 #8 The length of a rectangle is 4 inches less than twice the width. The perimeter is 112 inches. Find its dimensions. 1.6 #42 Solve for. 1.6 #48 Solve for h. 1.6 #10 A triangular garden has sides that can be represented b 3 consecutive integers. If the perimeter is 15 ft, what are the lengths of the sides? Solve for w. 11
13 1.6 #10 The smallest angle in a triangle is the size of the largest angle. The middle angle measures less than the largest. Find all three angle measures. 1.7 Solving Inequalities *Be careful to change the direction of the inequalit whenever ou or each side of the inequalit b a. *For each of the following, graph our solutions sets. Also write our solutions using interval notation. 1.7 # #20 Find : (+15) 1.7 #36 12
14 Ke Phraes: at most at least no more than no less than Aleks Problem: Brian is choosing between two eercise routines: In routine #1, he burns 46 calories walking. He then runs at a rate that burns 11 calories per minute. Aleks Problem: For her phone service, Kira pas a monthl fee of $20, and she pas an additional $0.04 per minute of use. The least she has been charged in a month is $ What are the possible numbers of minutes she has used her phone in a month? Use for the number of minutes, and solve our inequalit for. In routine #2, he burns 22 calories walking. He then runs at a rate that burns 15 calories per minute. For what amounts of time spent running will Routine #1 burn fewer calories than Routine #2? Use for the number of minutes spent running, and solve our inequalit for. 13
15 1.8 Eponents & Scientific Notation A. Properites of Eponents 1. *Simplif each of the following: a. b. 2. c. d. 3. e. f. 4. g. 5. h. i. j. Negative eponents are NOT considered to be simplified. Do NOT leave them in final answers! 14
16 k. s. l. m. 6. ALEKS PROPLEM TYPE: Simplif: n. o. p. q. r. Multiple choice: a. b. c. d. 15
17 B. Scientific Notation Scientific notationis a shorthand notation for writing etremel small or large numbers. *Multipl. Write our answers in scientific notation: 6. Notation: 7. *Write each using scientific notation: 1. 9,374, trillion *Divide. Write our answers in scientific notation: *Write each in standard form:
18 Additional Aleks Problems (Chapter 1) 1. The properties of addition are: [1] Commutative Propert [2] Associative Propert [3] Additive Identit Propert [4] Additive Inverse Propert 3. Dale rented a truck for one da. There was a base fee of $20.99, and there was an additional charge of 95 cents for each mile driven. Dale had to pa $ when he returned the truck. For how man miles did he drive the truck? For each equation below, indicate the propert that justifies the equation b filling in the bo with the appropriate number. 2. Consider the following properties of real numbers: [1] Commutative Propert of Addition [2] Associative Propert of Addition [3] Additive Identit Propert [4] Additive Inverse Propert [5] Distributive Propert [6] Commutative Propert of Multiplication [7] Associative Propert of Multiplication [8] Multiplicative Identit Propert [9] Multiplicative Inverse Propert [10] Multiplication Propert of Zero 4. Raina invested her savings in two investment funds. The amount she invested in Fund A was 3 times as much as the amount she invested in Fund B. Fund A returned a 4% profit and Fund B returned a 6% profit. How much did she invest in Fund A, if the total profit from the two funds together was $1080? For each equation below, indicate the propert that justifies the equation b filling in the bo with the appropriate number. 5. The price of an item has been reduced b 80%. The original price was $90. What is the price of the item now? 17
19 2.1 Graphing Lines Graph: Methods for Graphing Lines: 1. The XY Chart Graph: Graph: 2. Horizontal and Vertical Lines 4. Graphing b Intercepts Graph: 3. The SlopeIntercept Method A. Identif and plot the intercept. B. Identif the slope. Rise and run (to the right) from the intercept to another point on the line. C. Draw the line. 18
20 2.2 Slope Formula: *In each equation, identif the slope and the intercept. 3. *Find the slope of each line: *Find the slope of the line which runs through the given pair of points: *Find the slope of the graphed line (passes through and :
21 2.3 PointSlope Form **Parallel lines have the slope. **The slopes of perpendicular lines are of each other. *Find the equation of the line (in slopeintercept form) which has has the following characteristics. a. and has intercept *Are the following pairs of lines parallel, perpendicular, or neither? 1. b. and the line passes through Three Forms of a Line c. passes through and
22 d. passes through and is parallel to the line Aleks Problem: Consider the line. Find the equation of the line that is parallel to this line and passes through the point. Find the equation of the line that is perpendicular to this line and passes through the point. e. passes through and is perpendicular to the line Aleks Problem: Write equations for the horizontal and vertical lines passing through the point. horizontal line: vertical line: 21
23 2.4 Word Problems 2.4 #8 Ale is a sales representative and earns a base salar of $1000 per month plus a 4% commission on his sales for the month. a. Write a linear equation that epresses Ale s monthl salar in terms of his sales. 2.4 #14 Let represent the average number of miles driven per ear for passenger cars in the United States since Let =0 represent the ear where corresponds to 1980, =1 corresponds to 1981, and so on. The average earl mileage for passenger cars can be approimated b the equation = where. b. Graph the equation. a. Use the linear equation to approimate the average earl mileage for passenger cars in the United States in the ear c. What is the intercept and what does it represent in the contet of this problem? d. What is the slope of the line and what does it represent in the contet of this problem? b. Use the linear equation to approimate the average mileage for the ear 1985, and compare it with the actual value of 9700 mi. e. How much will Ale make if his sales for a given month are $30,000? c. What is the slope of the line and what does it mean in the contet of this problem? d. What is the intercept and what does it mean in the contet of this problem? 22
24 Aleks Problem: Owners of a recreation area are filling a small pond with water. The are adding water at a rate of 35 liters per minute. There are 700 liters in the pond to start. Aleks Problem: The monthl cost (in dollars) of a longdistance phone plan is a linear function of the total calling time (in minutes), as shown in the figure below. Let W represent the amount of water in the pond (in liters), and let T represent the number of minutes that water has been added. Write an equation relating W to T, and then graph our equation using the aes below. The monthl cost for 51 minutes of calls is $17.52 and the monthl cost for 79 minutes is $ What is the monthl cost for 69 minutes of calls? 23
25 Aleks Problem: Suppose that the credit remaining on a phone card (in dollars) is a linear function of the total calling time (in minutes). When graphed, the function gives a line with a slope of. See the figure below. There is $25.04 in credit remaining on the card after 42 minutes of calls. How much credit was there after 29 minutes of calls? 2.5 Domain and Range The domain of an epression is the set of values which substituted into the epression. The range of an epression is the set of values which can the epression. Eample: domain: range: To find the domain, start with a "default" domain of and then take awa values which... ** ** ** *Find the domain of each: 24
26 To determine the domain from a graph, look at where the graph etends, left to right. To determine the range from a graph, look at where the graph etends verticall. *Find the domain and range of each graph: *Find the domain and range of each: 25
27 2.6 Functions A function is a rule which assigns a value in the range to each value in its domain. *Do the following relations represent functions? When we write, we mean Given,, find the following: 1. A graph is that of a function if it passes the 2. *Are the following graphs those of functions? Notation: is pronounced Antime ou see., ou can replace it with 8. 26
28 *Find the requested values from the graph: 2.6 #57 The graph of is given. *Find the domain of each function. Write the answers in interval notation. 2.6 #64 a. Find. b. Find 2.6 #65 c. Find d. For what value(s) of is e. For what value(s) of is 2.6 #68 f. Write the domain of. g. Write the range of. 2.6 # #58 The graph of is given. a. Find. 2.6 #73 b. Find 2.6 #76 c. Find d. For what value(s) of is e. For what value(s) of is A. h ( ) 1 5 f. Write the domain of. g. Write the range of. 27
29 2.7 Graphs of Basic Functions Graph each of the following. Then state the domain and range of each. 4. Absolute Value Function 1. Linear Function 5. Square Root Function 2. Quadratic Function 6. Reciprocal Function 3. Cubic Function 28
30 Graph the parabola points on its graph.. Indicate five Review: Solve the following equation for : Graph the cubic function three points on its graph.. Indicate Simplif: Aleks Problem: For each function below, choose the correct description of its graph: 29
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