Math 002 Intermediate Algebra Fall 12 Objectives & Assignments

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1 Math 002 Intermediate Algebra Fall 12 Objectives & Assignments Unit 1 Equations in 1- & 2- Variables, Inequalities in 1- Variable, & Absolute Value I. Language and Operations with Algebra 1. Use the order of operations to simplify expressions. 2. Evaluate algebraic expressions. 3. Translate verbal statements into algebra. II. Equations in One Variable and Problem Solving 4. Solve a linear equation in one variable. 5. Identify a conditional equation, contradiction, and identity. 6. Solve a literal equation in one variable. 7. Solve applications: mark up/down, geometry, interest calculations, number relationships. III. Inequalities in One Variable 8. Solve inequalities and compound inequalities in one variable. 9. Graph the solution on a number line. 10. Write solutions in interval notation. 11. Solve applications that can be modeled by linear inequalities. IV. Absolute Value 12. Solve absolute value equations. v. Graphs and Functions 13. Graph equations in two variables by plotting points. 14. Graph linear equations by finding and plotting intercepts. 15. Use function notation. 16. Identify the domain, range, and function values from a graph. II. Lines and Linear Functions 17. Calculate the slope of a line through two points. 18. Graph lines using the slope and y-intercept. 19. Determine if two linear equations represent parallel or perpendicular lines. 20. Write equations for lines that a. pass through two given points. b. have a specified slope through a given point. c. are parallel or perpendicular to a given line through a specific point. d. are horizontal or vertical through a given point. 21. Solve applications involving linear relationships. 1

2 ASSIGNMENTS Unit 1 Type Homework Topic Due BOOK pg. 92: 48, 63, 72, 78 Introduction - Optional 1.5 BOOK pg. 50: 22, 32, 36, 42, 54 Solving Linear Equations MML Orientation & A1 (Rev) TH Aug BOOK pg. 60: 28, 44, 46, 64, 66 Applications 1.8 BOOK pg. 83: 10, 14, 28, 34, 38 Formulas & Problem Solving MML A2 ( ) M Aug BOOK pg. 227: 30, 50, 56, 66, 70, 76 Linear Inequalities 3.3 BOOK pg. 237: 18, 28, 42, 65, 70 Compound Inequalities 3.4 BOOK pg. 243: 6, 16, 40, 54,78, 85 Absolute Value Equations MML A3 (3.2, 3.3) MML Quiz 1 TH- Aug BOOK pg. 113: 30, 70, 72, 76, 80 Graphing 2.2 BOOK pg. 133: 34, 40, 45, 72, 78, 86, 88 Functions 2.3 BOOK pg. 146: 20, 40, 46, 50, 62, 72 Graphing Linear Functions MML A4 (3.4, 2.1, 2.2) TH Sept 6 MML A5 (2.3, 2.4) M-Sept BOOK pg. 160: 4, 26, 50, 62, 68 The Slope of a Line EXAM 1 Group A: TH- Sept 13 & Group B: FR Sept 14 Math 002 Unit 1 - Student Notes Section Review 1. Using the Order of Operations: PEMDAS; p. 93 # (M3) Complete #1 on Quick Check as an example. Parentheses Exponents Multiplication/Division from left to right Addition/subtraction from left to right 2. Evaluate an expression containing a variable(s): p. 93 #78 Find the value of when 2

3 Section Solving Linear Equations Algebraically Objectives Solve linear equations o using properties of equality. o that can be simplified by combining like terms. o containing fractions or decimals. Check solutions to linear equations. Identify conditional equations, identities, and contradictions. Vocabulary: Solution, equivalent equations, conditional equation, identity, contradiction 1. Linear equations in one variable. An example of a linear equation in one variable: Definition of a linear equation in one variable: 2. Solving linear equations: Solve the following equations and check the solution. general. EX 1, EX 3, EX 4 a) b) 3

4 3. Define solution: 4. Checking a solution by hand. EX 1 EX 7 using the graphing calculator. EX 1 EX 7 5. More types of equations. containing fractions. EX 5 c) containing decimals. EX 2, EX 7 d) 6. Summarize steps to solve algebraically: Clear fractions, decimals, and groupings (parentheses). Combine like terms on each side of the equation. Use the addition/ subtraction properties of equality to isolate the variable term. Use the multiplication/division properties of equality to solve for the variable. 4

5 7. Types of linear equations. The equations we have looked at so far have all been conditional equations, they are equations in which some specific number satisfies the equation and other numbers don t. Conditional - solution: EX 1 EX 7 Ex: A contradiction is an equation that is not true for of its variable. This is also called an impossible equation. Ex: Contradiction solution: EX 8 An identity is an equation that is true for of its variable. Ex: Identity solution: EX 9 Concept Check: I By inspection, decide which equations have no solution and which equations have all real numbers as solutions. Then identify the equation as a contradiction or identity. a) b) c) d) Solve. e) 3( x 5) 4( x 4) x 1 f) 4( c 3) 2c 2(6 c) 5

6 1.6 An introduction to Problem Solving Objectives: Write algebraic expressions that can be simplified. Apply the steps for problem solving. Writing and Simplifying Algebraic Expressions In order to prepare for problem solving, we practice writing algebraic expressions that can be simplified. 1. Writing algebraic expressions from verbal expressions. Identify key words/phrases: more/less than, increase/decrease, times, of, is, etc. Addition Subtraction Multiplication Division Consecutive integers, consecutive even integers, consecutive odd integers Consecutive Integers: Consecutive EVEN Integers: Consecutive ODD Integers: Example 1 Write the following as algebraic expressions. Then simplify. a) The sum of two consecutive even integers, if x is the first even integer. b) The perimeter of the rectangle with sides of length x and 2x General Strategy for Problem Solving: This is in their books. a. Understand the problem. Become comfortable with the problem: i. Read and reread the problem. ii. Construct a drawing. iii. Choose and define a variable to represent the unknown. b. Translate the problem into an equation c. Solve the equation d. Interpret the results: Check the proposed solution in the stated problem and state your conclusion. 6

7 3. Topics: Consecutive integers; EX 1a, EX 6 The sum of the first and third of three consecutive even integers is 156. Find the three even integers. Finding unknown numbers; EX 3 A second number is three times the first number. A third number is 50 more than the first number. If their sum is 235, find the three numbers. Perimeter; EX 1b; EX 5 In a blueprint of a rectangular room, the length is to be 2 centimeters greater than twice its width. Find the dimensions if the perimeter is to be 40 centimeters. Mark-up (increase) /mark-down (decrease) ; EX 4 Luisa is a computer programmer. Her new salary of $30,000 reflects a 20% raise. What was her salary before the raise? 7

8 1.8 Formulas and Problem Solving Objectives: Solve a formula for a specified variable. Use formulas to solve problems. 1. Define formula: A formula is an equation that. Some examples: Formula Meaning Area of a rectangle = length width Other formulas are listed in the front cover of your text. 2. Solve a formula for a specified variable. These problems are similar to solving a linear equation in one variable. EX 1, EX 2, EX 3 Example 1: Solve each equation for the specified variable. You might want to circle the term that includes the variable you are trying to isolate. #10 y mx b ; for x #14 ; for b #24 8

9 3. Use formulas to solve problems. Compound Interest: EX 4 You will not need to memorize this formula but you will need to recognize it, know what the variables stand for so that you can substitute appropriately and evaluate using the graphing calculator. A = value for n : P = r = t = n = annually semiannually monthly quarterly weekly daily #28 A principal of $25,000 is invested in an account paying an annual percentage rate of 5%. Find the amount in the account after 2 years if the account is compounded a) semiannually b) quarterly c) monthly n A Temperature; area; distance The formula converts degrees Fahrenheit to degrees Celsius. Use this formula and the table feature of your calculator to complete the given table. If necessary, round values to the nearest tenth. Fahrenheit x Celsius 9

10 3.2 Linear Inequalities (in one variable) and Problem Solving Objectives: Use interval notation. Solve linear inequalities using the addition and the multiplication properties of inequalities. Solve problems that can be modeled by linear inequalities. 1. Inequality symbols: < > 2. A solution of an inequality is a value that makes the inequality a true statement. The solution set of an inequality is the. 3. Representations of a solution set: EX 1 a. Set notation: b. Graph: c. Interval notation: Sketch the graph. d) e) f) 4. Solving linear inequalities is similar to solving linear equations. Addition/Subtraction property of Inequality: can add/subtract the same number to both sides of the inequality and the direction of the inequality symbol remains the same. EX 2, EX 3 10

11 Multiplication property of inequality: multiply/divide both sides i. by a positive number the direction of the inequality symbol. EX 4a, EX 5 ii. by a negative number the direction of the inequality symbol. EX 4b, EX 6 Check algebraically. (Skip check graphically) Example 1: Solve the simple linear inequalities, graph their solution sets, and write solution sets in interval notation. a) b) c) d) e) 5. Solve applications modeled by a linear inequality. EX 8, EX 9 Example 2: Use an inequality to solve the following application. Shureka Washburn has scores of 72, 67, 82, and 79 on her algebra tests. i) Use an inequality to find the scores she must make on her final exam to pass the course with an average of 77 or higher, given that the final exam counts as two tests. ii) In your own words explain the meaning of your answer to part a. 11

12 3.3 Compound Inequalities Objectives Find the intersection of two sets. Solve compound inequalities containing and. Find the union of two sets. Solve compound inequalities containing or. Reflect on these situations: A. You get a discount if you are at least 18 years old and no more than 60 years old. B. You get a discount if you are less that 18 years old or at least 60 years old. C. You get a discount if you are less than 18 years old and at least 60 years old. 1. Compound inequalities: Two inequalities joined by the words and or or. Two inequalities joined by the words and or or are called compound inequalities. Some examples are: 2x x 3 8 and x 2 5 or x Find the intersection of two sets; and; : The set of all elements where the sets overlap. EX 1 Find the intersection: The solution set of a compound inequality formed by the word and is the of the solution sets of the two inequalities. 3. Solve compound inequalities containing and. For the following examples solve each compound inequality, graph its solution set and IB in interval notation. ** Don t forget to check the endpoints whether to include them or not in the solution set. a. one-sided interval. EX 2 and 12

13 b. contained interval in between. EX 4, EX 5 and c. Compact form. EX You can solve a compound inequality written in compact form by by applying transformations to each member of the inequality. i) ii) d. No solution. EX 3 and 4. Find the union of two sets; or : The set of elements that belong to either of the sets. EX 6 The union of two sets A and B, is the set of all elements. In either set A or B or : union: Find the union: 13

14 5. Solve compound inequalities containing or. The solution set of a compound inequality formed by the word or is the union of the solution sets of the two inequalities. Solve each compound inequality, graph its solution set and write it in interval notation. ** Don t forget to check the endpoints whether to include them or not in the solution set. a. Two separate intervals. EX 7 b. One interval. #33 c. All real numbers. EX 8 6. The importance of order a. in an inequality: What is wrong with? b. in an interval: What is wrong with? 14

15 3.4 Absolute Value Equations Objectives: Understand the geometric representation of the absolute value. Solve absolute value equations 1. : The absolute value of a number is its on the real number line. means that is units from zero. Where is? Either at or. a. Verbal phrase: means that b. Graph: is units from zero. c. Equality: Where is? Either at or. Verbal Phrase The distance from x to 0 is 4. Absolute Value Eqn Graph Solution 2. Solving absolute value equations of the form where. Set up two equations EX 1, EX 2, EX 3 SOLVING EQUATIONS OF THE FORM X a If a is a positive number, then X a, where, is equivalent to a. b. 15

16 of the form where. EX 4 of the form where. EX 7 Of the form. EX 8 16

17 Section 2.1- Graphing Equations 3. Review Cartesian coordinate system: x axis, y axis, quadrants, plotting ordered pairs x coordinate, y coordinate. EX 1 * Note: An ordered pair is given in alphabetical order. Label the following: x- and y- axes origin Quadrants Points: A(3, -2) B(0, 3) C(-4, 1) D(-1, 0) E(-2.5, -3) F(3.5, 4.5) 4. Linear equations in two variables. Some examples: Standard Form: Example 1: Look at the following graph of. Pick a point on the graph and identify the coordinates. What does it mean for a point to be on the graph? 5. Define a solution of an equation in two variables. EX 3 solution: Example 2: Determine whether lies on the graph of. What about (5, 105)? 17

18 6. Graphing linear and nonlinear equations by hand. EX 5, EX 6, EX 8, EX 10 a. For a linear equation: Find 3 ordered pairs; plot these solutions; the line through the plotted points is the graph. b. For a nonlinear equation: Find many ordered pairs; plot the solutions; connect the plotted points with a smooth curve to sketch its graph. ; Let. x y ; Let. x y ; Let. x y 7. Graphing linear and nonlinear equations using a graphing calculator. EX 5, EX 7, EX 8, EX 9, EX 10 a. Get equation in the form y = expression. b. Enter in calculator. c. You can change the window to get an accurate picture of the graph. d. You can use the table feature to see a list of ordered pairs. 18

19 Section Introduction to Functions 8. Define relation, domain, and range. EX 1 A relation is. The following correspondences describe a relation. a) b) c) Input States Arkansas Kentucky South Carolina Texas Output # of Reps The domain of a relation is the set of all of the ordered pairs. The range of a relation is the set of all of the ordered pairs. Example 1: Determine the domain and range of each relation. 9. Define function: A relation in which each first component corresponds to exactly. Some examples: Determine whether a relation is a function. EX 2, EX 3 In Example 1 above, which relations are also functions? Use the vertical line test to determine whether a relation is a function. EX 5, EX 6 vertical line test: 10. Find the domain and range of a function. EX 6 Example 2: Find the domain and range of each relation. Determine whether the relation is also a function. 19

20 11. Use function notation. EX 7, EX 8 To denote that is a function of, we can write: What does this mean? o Dependent variable: o Independent variable Example 3: If and, find: a. b. What are the ordered pairs that correspond to these values? 12. Use a graph to identify function values, domain, and range. EX 9, EX 11 Example 4: Consider the graph of Q(x). a. Find. b. Find. c. Find. d. For what values of is e. Identify the domain: f. Identify the range: 20

21 Section 2.3 Graphing Linear Functions 4. Review graphing linear equations by making a table of values, plotting ordered pairs, and connecting with a line. Section Define and find - and - intercepts. EX 4, EX 5, EX 6 Example 1: Identify where the following graphs cross the -axis and -axis. Label the points. a) b) o - intercept: point where the graph crosses the -axis. To find an x- intercept, let or and solve for x. o - intercept: point where the graph crosses the -axis. To find a y- intercept, let and solve for y o Remember these are points. Write as an ordered pair. 6. Graphing linear equations using - and - intercepts. EX 4 Example 2: Graph the lines using the - and -intercepts. a) b) c) 7. Graphing vertical and horizontal lines. EX 7, EX 8 o Horizontal lines: o Vertical lines: 21

22 Section 2.4 The Slope of a Line 1. Find the slope of a line given two points on the line. EX 1, EX 2 Plot points (2, 1) and (4, 5). Sometimes you can look at the graph to calculate the slope of the line, but sometimes it can be difficult. Slope of a line: Given a line passing through the points, the slope m of the line is: Example 1: Find the slope of line that passes through the points and. Summarize the overall appearance of lines with positive, negative, zero, and undefined slope. H O Y V U X : EX 5, EX 6 2. Different forms of an equation of a line. a. Standard form: b. Slope-intercept form: c. Function notation: 3. Find the slope of a line given the equation of a line. EX 3 Write the equation in slope-intercept form:. Example 2: Find the slope and -intercept of each line. Then graph each line. a) b) c) 22

23 4. Slopes of parallel and perpendicular lines. EX 7 a. Two non-vertical lines are parallel if they have b. Two non-vertical lines are perpendicular if they have Example 3: The slope of a line is. Slope of a parallel line: Slope of a perpendicular line: Example 4: Given the line, (a) Give an equation of a line parallel to the given line: (b) Give and equation of a line perpendicular to the given line: Example 4: Determine whether the lines are parallel, perpendicular or neither. a) f ( x) 3x 6 g( x) 3x 5 b) 2x 3y 1 3x 2y 12 23

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