California State University, Northridge


 Cora Kennedy
 2 years ago
 Views:
Transcription
1 California State Universit, Northridge MATH 09 HYBRID WORKBOOKS Spring 00
2 Chapter Equations, Inequalities and Applications. The Addition Propert of Equalit Learning Objectives:. Use the Addition Propert of Equalit to solve linear equations.. Simplif an equation and then use the Addition Propert of Equalit.. Write word phrases as algebraic epressions.. Ke Vocabular: solving, equivalent equations, addition propert of equalit. A. Using the Addition Propert Definitions:. Linear Equation in One Variable is an equation of the form A B C, where A, B and C are an real numbers and A 0.. Addition Propert of Equalit: If a b, then a c b c, where a, b and c are an real numbers.. Distributive Propert: a ( b c) ab ac and a( b c) ab ac, where a, b and c are an real numbers. Eample. Solve each equation. Check each solution.. 8 t. a B. Simplifing Equations Steps to Simplif Equations:. Simplif each sides of equation as much as possible.. If an equation contains parentheses, use the distributive propert to remove the parentheses.. Using the proper of equalit to solve the resulted equation.
3 Eample. Solve each equation a.7 a. 7a ( ) ( )
4 C. Writing Algebraic Epressions Algebraic Epressions are epressions that contain variable. Eample. Write each algebraic epression described.. Two numbers have a sum of 7. If one number is, epresses the other number in terms of.. A 6 foot board is cut into two pieces. If one piece is feet long, epress the other length in terms of.. On a recent car trip, Ramond drove miles on da one. On da two, he drove 70 miles more than he did on da one. How man miles, in terms of, did Ramond drive for both das combined?. Eercise Solve each equation r.. 7. k p 7 0p
5 ( 6k) 7k. ( ). ( 9) 6. 7 ( z) 8( z ) z o 7. Two angles have a sum of6. If one angle is o o, epress the other angle in terms of. 8. A foot board is cut into two pieces. If one piece is feet long, epress the other length in terms of. 9. From Chicago, it is more miles to Montreal than it is to New York Cit. If it is m miles to New York, epress the distance to Montreal in terms of m.
6 . The Multiplication Propert of Equalit Learning Objectives:. Use the multiplication propert of equalit to solve linear equations.. Use both the addition and multiplication properties of equalit to solve linear equations.. Write word phrases as algebraic epressions.. Ke Vocabular: reciprocal, consecutive integers. A. Using the Multiplication Propert Multiplication Propert of Equalit: If a b then ac bc where a, b and c are an real numbers. Eample. Solve the following linear equations ( ) 8 ( ) 6
7 B. Writing Algebraic Epressions Eample. Write each algebraic epression described. Simplif if possible.. If represents the first of two consecutive even integers, epress the sum of the two integers in terms of.. If represents the first of three consecutive odd integers, epress the sum of the first and third integer in terms of.. Eercise Solve each equation d p k 9. b.. 7 z 8 z z 8z z 7 6z 8. If is the first of three consecutive even integers, write their sum as an algebraic epression in. 9. Houses on one side of a street are all numbered using consecutive odd integers. If the first house on the street is numbered, write an epression in term of for the sum of five house numbers in a row. 7 7
8 . Further Solving Linear Equations Learning Objectives:. Appl the general strateg for solving a linear equation.. Solve equations containing fractions and decimals. Recognize identities and equations with no solution.. Ke Vocabular: least common denominator (LCD), identit, no solution. Steps for Solving Linear Equations. If an equation contains fractions, multipl both sides b the LCD to clear fractions.. If an equation contains decimal, multipl both sides b the power of ten according to the numbers of the decimal digit.. If and equation contains parentheses, use distributive propert to remove the parentheses.. Simplif each side of the equation b combining like terms.. Get all variable terms on one side and all numbers on the other side b using the addition or the multiplication propert of equalit. 6. Check the solution b substituting the result into the original equation. Eample. Solve the following linear equations.. 6 a ( a ). ( ) ( ). 6 8
9 ( 00) Eercise Solve each equation ( 7) 0. ( n ) ( 7n ) 8. 6( ) ( ). 7 ( ) ( 8) 6. ( ) ( 9 a ) a a k k 6 ( ) ( ) ( 80) 0.( 68). 0.( 00) ( 00). 0 ( ) ( 0) ( 0) 8. 9( ) 6( 6 ) 9. The perimeter of a geometric figure is the sum of the lengths of its sides. If the perimeter of a trapezoid is 9 cm, and the length of the sides are,, ( ) and cm, find the length of each side. 9
10 . An Introduction to Problem Solving Learning Objectives:. Translate a problem to an equation, and then use the equation to solve the problem.. Ke Vocabular: understand, translate, solve, and interpret. General Strateg for Problem Solving. Understand the problem b doing the following Read and reread the problem carefull. Choose a variable to represent the unknown quantities. Construct a drawing if needed.. Translate the problem into an equation.. Solve the equation using algebra.. Verif the solution (Check if the answer making sense). Eample. Solve each word problem.. Eight is added to a number and the sum is doubled, the result is less than the number. Find the number.. The difference between two positive integers is. One integer is three times as great as the other. Find the integers.. When ou open a book, the left and right page numbers are two consecutive natural numbers. The sum of their page numbers is 9. What is the number of the page that comes first? 0
11 . A college graduating class is made up of 0 students. There are 06 more girls than bos. How man bos and girls are in the class?. A ft pipe is cut into two pieces. The shorter piece is 7 feet shorter than the longer piece. What is the length of the longer piece? 6. A triangle has three angles, A, B, and C. Angle C is 8 greater than angle B. Angle A is times angle B. What is the measure of each angle? (Hint: The sum of the angles of a triangle is 80 ).. Eercise Solve each word problem.. The sum of five times a number and 8 7 is equal to the difference between si times a number and. Find the number.. Eight times the sum of a number and is the same as nine times the number. Find the number.
12 . Twice the difference of a number and seven is equal to five times the number plus one. Find the number.. If the sum of a number and is doubled the result is times the number. Find the number. 7. Sue makes twice as much mone as Tom. If the total of their salaries is $78,000, find the salar of each. 8. Pegg Fleming won two more U.S. Figure Skating Championships than Doroth Hamill. If the total championship for both is 8, find how man each won. 9. A 0 inch board is to be cut into three pieces so the second piece is twice as long as the first piece and the third piece is three times as long as the first piece. Find the length of all three pieces. 0. A 6 inch board is to be cut into three pieces so the second piece is three times as long as the first piece and the third piece is four times as long as the first piece. Find the length of all three pieces.. A carpenter gave an estimate of $980 to build a cover over a patio. His hourl rate is $8 and he epects to need $60 in materials. How man hours does he epect the job to take?. A mechanic charged $9 to repair a car, including $07 in parts and 6 hours of labor. How much does she charge per hour for labor?. An appliance repairman charges $7 to come to our house and $ per hour. During one week, he visited 9 homes and his total weekl income was $0. How man hours did he spend working on appliances?. Two angles are supplementar if their sum is 80. One angle measures four times the measure of an angle supplementar to it. Find the measure of the angle.. Two angles are complementar. The second angle is si less than three times the first. Find the two angles. 6. The height of a soup can is. cm more than its diameter. If the sum of the height and the diameter is 6., find each dimension. 7. Find two consecutive even integers so that three times the smaller is 0 more than two times the larger. 8. Find three consecutive odd integers whose sum is negative Karl s license plate is four consecutive integers with a sum of 6. What is his license plate number? 0. The sum of the angles of an four sided polgon is 60. If the measures of the angles of a four sided polgon are four consecutive odd integers, find the measure of each angle.
13 . Formulas and Problem Solving Learning Objectives:. Given a formula and values, solve for the unknown.. Solve a formula or equation for one of its variables.. Solve word problems.. Ke Vocabular: formula, perimeter, area, volume. A. Using Formulas to Solve Problems Formula describes a known relationship among quantities. Eample. Substitute the given values into each given formula and solve for the unknown variable.. Distance Formula: d rt ; t 9, d 6. Volume of a pramid: V Bh; V 0, h 8 B. Solving a Formula for a Variable Steps for Solving Equations for a Specified Variable. Multipl both sides of equation to clear fractions if the occur.. Use the distributive proper to remove parentheses if the occur.. Simplif each side of the equation b combining like terms if needed.. Get all terms containing the specified variable on one side and all other terms on the other side b using the addition propert of equalit.. Get the specified variable alone b using the multiplication propert of equalit. Eample. Solve each formula for the specified variable.. A bh for b.
14 . L d π ( a r) for a. Eample. Solve. Convert the record high temperature of 0 F to Celsius. (Use the formula F 9 C ). You have decided to fence an area of our backard for our dog. The length of the area is meter less than twice the width. If the perimeter of the area is 70 meters, find the length and width of the rectangular area.. Eercise Substitute the given values into the formula and solve for the unknown variable.. D rt when D 7 and r 68. V π r h when V 7. and r (Leave the answer in term of π.)
15 . V r π when r (Leave the answer in term of π.). A ( B b)h when A., B 8 and h 7.. A ( B b)h when A 77, B and b m m when,, and 8 when,, 7 m and Solve the following applications. 8. Wade has 6 inches of inch wide bias tape for a border on a rectangular banner. If the banner needs to be 8 inches long, what is the maimum width it could be? 9. It is 8 miles from Gumon to Tulsa. How long should it take Manuella to drive from Gumon to Tulsa if she averages driving 0 miles per hour? Use the formula d rt The formula F C can be used to convert temperatures in degrees Celsius to degree Fahrenheit. Convert Istanbul, Turke s 8 C average dail high in Jul to Fahrenheit.. Find how man piranhas ou can put in a clindrical tank whose diameter is feet and whose height is. feet if each piranha needs. cubic feet of water.. Which has more pizza, one 0 inch pizza or two inch pizzas, if the size indicates the diameter of a round pizza? Solve each formula for the specified variable.. P a b c d for d. 7 0 for. V Ah for A
16 .6 Solving Linear Inequalities Learning Objectives:. Graph inequalities on a number line.. Use the addition propert of inequalit to solve inequalities.. Use the multiplication propert of inequalit to solve inequalities.. Use both properties to solve inequalities.. Solve problems modeled b inequalities. 6. Ke Vocabular: inequalit, <, <, >, >, addition propert of inequalit, multiplication propert of inequalit, at least, no less than, at most, no more than, is less than, is greater than. A. Graphing Inequalities on a Number Line Inequalit is a statement that contains <, <, >, > smbols. Eample.. Graph each inequalit on a number line m < t B. Solving the Inequalities using the Addition and Multiplication Propert of Inequalit Properties of Inequalities Let a, b and c be real numbers, then. Addition Propert: If a < b, then a c < b c and If a > b, then a c > b c.. Positive Multiplication Propert: (c is positive) If a < b, then ac < bc and If a > b, then ac > bc.. Negative Multiplication Propert: (c is negative) If a < b, then ac > bc and If a > b, then ac < bc. 6
17 CAUTION! If multipl or divide b a negative number, the inequalit sign change to opposite. Eample. Solve each inequalit. Graph the solution set.. a > a ( ) ( ) > ( )
18 C. Solving Applications Involving Inequalities Ke words: Is less than means < At most means Is greater than means > At least means No more than means Not equal to means Is less than or equal to means Is greater than or equal to means Eample. Solve the following.. Eight more than twice a number is less than negative twelve. Find all numbers that make this statement true.. One side of a triangle is si times as long as another sides, and the third side is 8 inches long. If the perimeter can be no more than 06 inches, find the maimum lengths of the other two sides..6 Eercise Graph each on a number line.. >. Solve each inequalit.. < < 6 6. < >
19 0. < ( ). ( 7 6) < ( ). 8 ( ) ( ). 7 ( ) ( ). ( 8) ( ). 7( ) > ( ) 6 6. ( ) < ( ) Solve the following 7. Nine more than four times a number is greater than negative fourteen. Find all numbers that make this statement true. 8. Miranda needs an average of at least 90 to get an A in a course. She has earned scores of 8, 87 and 9 on her tests. The final eam counts as two tests. What score does she need on the final to get an A? 9. Tamara scored an 86 and a 9 on her last two math eams. What must she score on her third eam to have an average of at least a 9? 0. Ale has at most 90 ards of fencing available to enclose a rectangular garden. If the width of the garden is to be ards, find the maimum length that the garden can be. 9
20 Chapter Graphs and Linear Functions. Linear Equations and Their Solutions Learning Objectives:. Plot ordered pairs of numbers on the rectangular coordinate sstem.. Graph paired data to create a scatter diagram.. Find the missing coordinate of an ordered pair solution, given one coordinate of the pair.. Ke Vocabular: ordered pair, origin, quadrant, ais, ais, rectangular coordinate sstem, coordinate plane, coordinate, coordinate, paired data, scatter diagram, solution of an equation in two variables. Linear Equation is an equation of the form a b c or m b, where a, b, and c are an real numbers. m is the slope and ( 0, b) is the intercept. Solutions of Equations is an ordered pair (, ) that satisfies the given equation meaning when substitute the given ordered pair into the given equation will result a true statement. Eample. Complete each ordered pair so that it is a solution of the given linear equation.. 6 ; (, ). ;, Eample. Complete the table for the equation
21 . Eercise Complete each ordered pair so that it is a solution of the given linear equation.. 7 ; (, ). ;, 6 Complete the table of values for each given linear equation
22 . Graphing Linear Equations b Plotting Points Learning Objectives:. Graph a linear equation b finding and plotting ordered pair solutions.. Graph a linear equation and use the equation to make predictions.. Ke Vocabular: linear equation in two variables, graph of the equation, horizontal line, vertical line. Eample.. 8 For each equation, find three ordered pair solutions. Then use the ordered pairs to graph the equation...
23 .. Eercise For each equation, find three ordered pair solutions b completing the table. Then, use the ordered pairs to graph the equation Graph each linear equation. Label at least three points on the graph grid Write the statement as an equation in two variables. Then graph the equation. 0. The value is 6 less than the value.. The sum of and is 7.
24 . Graphing Lines Using Intercepts Learning Objectives:. Identif intercepts of a graph.. Graph a linear equation b finding and plotting intercept points.. Identif and graph vertical and horizontal lines.. Ke Vocabular: intercept, intercept, vertical line, horizontal line. A. Graphing Lines Using Intercepts. The intercept of a line is the point where the graph crossing the ais. To find the intercept Let 0, then solve for. Ordered pair for intercept: ( a, 0). The intercept of a line is the point where the graph crossing the ais. To find the intercept 0, b Let 0, then solve for. Ordered pair for intercept: ( ) Steps to Graph a Line Using the Intercepts.. Find the intercept ( a, 0).. Find the intercept ( 0, b)., 0 0, b, then connect them with a line.. Graph the points ( a ) and ( ) Eample. Graph and label at least two points on the graph grid
25 B. Graphing Vertical and Horizontal Lines. The Graph of a. The Graph of b is a vertical line with intercept (, 0) a. is a horizontal line with intercept (, b) 0. Eample. Graph and label at least two points on the graph grid Eercise Identif the intercepts and intercepts points..
26 ... Graph the line with intercept at 8 and intercept at 6. Graph each linear equation b finding and intercepts. Label the and intercepts on the graph grid Two lines in the same plane that do not intercept are called parallel lines. Graph the line. Then graph a line parallel to the line that intersects the ais at. What is the equation of this line? 6
27 . The Slope of a Line Learning Objectives:. Find the slope of a line given two points of the line.. Find the slope of a line given its equation including horizontal and vertical lines.. Compare the slopes of parallel and perpendicular lines.. Slope as a rate of change.. Ke Vocabular: slope, rise, run, zero slope, undefined slope. A. The Slope of Two Points The slope m if the line going through the points (, ) and (, ) b where is given B. The Slopes of Vertical and Horizontal Lines.. Vertical line a has.. Horizontal line b has. Eample. Find the slope of the line going through. (, ) and (,).. (,) and (,).. (, ) and (,). 7
28 C. The Slopes of Equations m b Slope of m b is. The intercept is. Steps of Finding a Slope from the Equation:. Write the given equation in the form.. Identif the slope and intercept. Eample. Find the slopes and the intercept of the following lines... 6 D. Finding Parallel and Perpendicular Lines. Parallel Lines Two lines are parallel if m m but b b.. Perpendicular Lines Two lines are Perpendicular if ( m )( m ) or m or m m m Eample. Decide whether the pair of lines is parallel, perpendicular or neither.. 8 8
29 . 6. Eercise Find the slope of each equation.... Find the slope of the line that goes through the given points.. (, ) and (, ). ( 7, ) and ( 7, 6) 6. ( 8, ) and (, ) 7. ( 6, 9) and (7, 0) 9
30 Find the slope of each line Determine whether the lines are parallel, perpendicular, or neither Use the points given, (a) find the slope of the line parallel and (b) find the slope of the line perpendicular to the line through each pair of points.. ( 8, ) and (, 6) 6. (, 6) and ( 7, 9) 0
31 . Equations of Lines Learning Objectives. Use the slope intercept form to write an equation of a line.. Use the slope intercept form to graph a linear equation.. Use the point slope form to find an equation of a line given its slope and a point on the line.. Use the point slope form to find an equation of a line given two points on the line.. Use the point slope form to solve word problems. 6. Ke Vocabular: standard form, slope intercept form, point slope form. A. Equations of Lines. Standard Form: A B C. Slope intercept Form: m b. Point Slope Form: m( ). Horizontal Line: b ; 0. Vertical Line: a ; m slope; ( 0,b) intercept ; (, ) given point; m slope m, ( 0,b) intercept; intercept none ; m undefined; ( a, 0) intercept; intercept none Eample. Find the slope intercept equation of the line that goes through the point (, ) and has a slope m, then graph. Eample. Find the slope intercept equation of the line having slope and intercept, then graph.
32 B. Finding the equation of lines given two points Steps for finding the slope intercept equation of a line given two points. Find the slope.. Find the equation of the line b first using the point slope form, and then write the equation in the form of m b... To graph, plot the given points ( ),, ( ), and joint them with a line. Eample. Find the slope intercept equation of the line going through the points, and,, then graph. ( ) ( ). Ecise Write the equation of each line in the form m b 7. m, b. m, b. m, b 0. m 0, b 9 8 Use the slope intercept form to graph each equation. Label at least two points on the graph grid Find the slope intercept equation of the line with given slope and passing through the given point. 9. m 9, through (, ) 0. m 7, through (, 6). m, through ( 6, ) 8 Find the equation of the line passing through each pair of points. Write the equation in the form A B C, 7,, 7, 9, 0, 0. ( ) and ( ). ( ) and ( ). ( ) and ( )
33 A certain tpe of notebook earned a stationar compan $,000 in profit the first ear and $,000 the third ear.. Assume the relationship between ears on the market and profit is linear. Use ordered pairs of t, ears on the market, and p, profit to write an equation of the relationship. 6. Use the equation to predict the profit the fifth ear.
34 .6 Introduction to Functions Learning Objectives:. Identif relations, domains, and ranges.. Identif functions.. Use the vertical line test.. Use function notation.. Ke Vocabular: relation, domain, range, function, vertical line test, function notation. A. Identifing Relations, Domains and Ranges: Definition: Relation is a set of ordered pairs. Domain of the relation is the set of all possible values. Range of the relation is the set of all possible values. Tpes of Functions. Linear Function: m b ; Domain: all real numbers; Range: all real numbers. Quadratic Function: a b c ; Domain: all real numbers. Rational Function: P ; Q Domain: all real numbers ecept Q 0 Range: all real numbers ecept 0 Eample. Find the domain and range:. T {( 6, ),(, 6),(, ) }. (, ) {, }. ( )
35 B. Identifing Functions Function is a set of ordered pairs in which each domain value has eactl one range value; that is, no two different ordered pairs have the same first coordinate. Function Notation: f ( ) read f of or f evaluate at Eample. Determine whether the relations are functions:. T {( 6, ),(, 6),(,) }. T {( 6, ),(, 6),(, ), (,) } {, }. ( ) {, }. ( ) {, }. ( ) C. Using the Vertical Line Test Vertical Line Test if a vertical line can be drawn so that it intersects a graph more than once, then graph is not the graph of a function. Eample. Determine whether the graph is that of a function...
36 D. Evaluating Functions Eample. Let f ( ). f ( ), find. f ( ) f ( ).6 Eercise Find the domain and range of each relation.. {(9, ), (0, 8), (, ), (, )}. {(7, 7), (, 7), (, 7)} Determine which relations are also functions.. {(9, ), (0, 8), (, ), (, )}. {(, ), (, ), (, )} Use vertical line test to determine whether each graph is the graph of a function
37 Given the function f ( ) 9, find the indicated function values. 7. f ( ) 8. f ( ) 9. f ( 0) Given the function f ( ) 7, find the indicated function values. 0. f () 0. f ( ). f ( ) 7
38 .7 Graphing Linear Inequalities in Two Variables Learning Objectives:. Determine whether an ordered pair is a solution of a linear inequalit in two variables.. Graph a linear inequalit in two variables.. Ke Vocabular: linear inequalit in two variables, half planes, boundar line. Linear Inequalit is an equation of the form: A B < C ; A B > C ; A B C ; A B C A. Graphing Linear Inequalities in Two Variables Steps to graph linear inequalities.. Solve inequalit in the form > m b.. If inequalit involving or, draws a solid line. If inequalit involving < or >, draws a dashed line.. Pick a test point. Substitute the values in the inequalit. If the result is true, shade the side that contain the test point. If a false statement, shade the other side. CAUTION! If multipl or divide b a negative number, the inequalit sign change to opposite. Eample. Graph the following inequalit and label at least two points on the graph grid
39 . < 0.7 Eercise Determine which ordered pairs are solutions of the linear inequalit. (, ). (, ). (, ) Graph each inequalit and label at least two points on the graph grid.. >. <
40 Chapter Sstems of Linear Equations. Solving Sstems of Equations b Graphing Learning Objectives:. Decide whether an ordered pair is a solution of a sstem of linear equations.. Solve a sstem of linear equations b graphing.. Identif special sstems: those with no solution and those with an infinite number of solutions.. Ke Vocabular: sstem of linear equations, parallel lines, no solution, infinite number of solutions, inconsistent sstem, consistent sstem, dependent equation, independent equations. A. Deciding Whether an Ordered Pair Is a Solution Sstem of Equations consists at least two or more linear equations. z Eample... z 0 z 0 Solution of the sstem is the point(s) where the graphs intersect. Eample. Is (,9) a solution of? Three tpes of the Sstem of Equations.. Consistent Sstem,. Two lines intersect at one point ( ) Has one solution (, ). m m When solve the sstem, get a number, a number.. Inconsistent Sstem Two lines are parallel. Has no solution. m m and b b When solve, get false statement. 0
41 . Dependent Sstem Two lines lie on top of the others (same line). Has infinitel man solutions. m m and b b When solve the sstem, get true statement. B. Solving Sstems of Equations b Graphing Steps for solving linear sstem b graphing.. Solve and graph each equation separatel.. Identif tpe of sstems (consistent, inconsistent, or dependent).. State number of solution (one solution, infinitel man solutions or no solution). Eample. Solve, graph, label tpe of sstem and state number of solution
42 . 8. Eercise Determine whether the ordered pair satisfies the sstem of linear equations.. ( ),. 0 6 ( ), ( ) 6, Solve each sstem of equations b graphing. State the solution(s) and tpe of sstem
43 . Solving Sstems of Linear Equations b Substitution Learning Objectives:. Use the substitution method to solve a sstem of linear equations.. Ke Vocabular: substitution method. Steps to solve linear sstem b substitution:. Solve one of the equations for one of its variable: or.. Substitute the resulting found in step into the other equation.. Solve the equation found in step to find the value of one variable.. Substitute the value found step in an original equations containing both variables to find the value of the other variable.. Check the solution b substituting the numerical values of the variables in both original equations. Eample. Solve, label tpe of sstem and state number of solution
44 . Eercise Solve each sstem of equations b substitution. State the solution(s) and tpe of sstem
45 . Solving Sstems of Linear Equations b Addition Method Learning Objectives:. Use the addition method to solve a sstem of linear equations.. Ke Vocabular: addition method, elimination method, opposite. Steps to solve a sstem of two linear equations b the addition method:. Rewrite each equation in standard form: A B C.. If necessar, multipl one or both equations b a nonzero number so that the coefficients of a chosen variable in the sstem are opposites.. Add both equations.. Find the value of one variable b solving the resulting equation from step.. Find the value of the second variable b substituting the value found in step into either one of the original equations. 6. Check the solution b substituting the numerical values of the variables in both original equations. Eample. Solve, label tpe of sstem and state number of solution
46 6. 6. Eercise Solve each sstem of equations b addition. State the solution(s) and tpe of sstem b a b a
47 . Applications of Sstem of Linear Equations Learning Objectives:. Use a sstem of equations to solve problems. Problem Solving Steps:. UNDERSTAND the problem b do the following: Read and reread the problem. Identif what is given and what is the question. Choose two variables to represent the two unknowns being asked. Construct a drawing if needed.. TRANSLATE the problem into two equations.. SOLVE the sstem of equations.. INTERPRET the results: Check the proposed solution in the stated problem and state our conclusion. A. Finding Unknown Numbers Eample. The sum of two numbers is 6. Their difference is. What are the numbers? B. Solving a Problem about Prices Formula: Number of tickets price per ticket total price Eample. Admission prices at a local weekend fair were $ for children and $7 for adults. The total mone collected was $79, and 87 people attended the fair. How man children and how man adults attended the fair? Numbers of tickets Price per ticket Total price children adults 7
48 C. Coin Problems Total Value numbers of coins value of each coin Eample. Tim has $.0 in quarters and nickels. How man quarters and nickels does she have if he has coins in total? Numbers of coins Value of each coin Total value quarters nickels D. Investment Problems I Pr t Where I interest earn, P principal, r interest rate, t time (in ear) Eample. Lit invested $6000, part at 6% and the rest at %. How much is invested at each rate if the annual income from the two investments is $90? P r t I Account Account 8
49 E. Miture Problems Formula: Amount of solution number of liters percent of the solution Eample. A pharmacist wants to make 0 liters of a 60% alcohol solution. She currentl has a 0% alcohol solution and a 70% alcohol solution. How man liters of a 0% alcohol solution and a 70% alcohol solution she needs to make 0 liters of a 60% alcohol solution? Number of liters Percent of solution Amount of solution Solution Solution Miture F. Geometr Problems Eample 6. The perimeter of a rectangle is 8 inches. The length is more than three times the width. Find the length and the width. 9
50 . Eercise Solve each problem using sstems of equations.. The sum of two numbers is 6. Their difference is. Find the two numbers.. The sum of two numbers is. The second number is more than twice the first. Find the numbers.. The difference between two numbers is 6. Five times the smaller is the same as 8 less than twice the larger. Find the numbers.. Two records and three tapes cost $. Three records and two tapes cost $9. Find the cost of each record and tape.. At school, two photograph packages are available. Package A contains class picture and 0 wallet size pictures for $9. Package B contains class pictures and wallet size pictures for $. Find the cost of a class picture and the cost of a wallet size picture. 6. A broker invested a total of $00 in two different stocks. One stock earned 9% per ear. The other earned 6% per ear. If $60 was earned from the investment, how much mone was invested in each? 7. The price of admission for a concert was $9 for adults and $ for children. Altogether, 770 tickets were sold, and the resulting revenue was $,680. How man adults and how man children attended the concert? 8. A druggist has one solution that is 0% iodine and another solution that is 0% iodine. How much of each solution should the druggist use to get 00 ml of a miture that is 0% iodine? 9. A chemist has one solution that is 0% alcohol and another that is 60% alcohol. How much of each solution should the chemist use to get 00 ml of a solution that is % alcohol? 0. The perimeter of a rectangle is cm. Two times the height is cm more than the base. Find the length of the height and length of the base.. The sum of the legs of a right triangle is 7 inches. The longer leg is more than twice the shorter. The hpotenuse is in. Find the length of each leg.. Two angles are complementar. The larger angle is 6 less than times the smaller angle. Find the measure of each angle.. Todd has 7 total coins in his bank, all dimes and quarters. The coins have a total value of $.9. How man of each coin does he have?. Nar has $.0 in dimes and nickels. She has 6 coins in total. How man dimes and nickels does she have? 0
51 Chapter Eponents and Polnomials. EXPONENTS Learning Objectives:. Evaluate eponential epressions.. Use the product rule for eponents.. Use the power rule for eponents, products, and quotients.. Use the quotient rule for eponents, and define a number raised to the 0 power.. Decide which rule(s) to use to simplif an epression. 6. Ke Vocabular: eponential epression, power, raised, product rule, same base, simplifing an eponential epression, power rules, quotient rule, zero eponent. Eponential Epression is epression of the form: a n a a a a, where a is the based, n is the eponent. n times Eponential Properties. If m and n are integers, and and are an real number, 0, 0, then. m n m. ( ) n. ( ) m m. m. n ( ) 0 m m m m m CAUTION! ( ) and ( ) m Eample. Evaluate each epression.. ( 7). 6.
52 when Eample. Use the properties of the eponent to simplif. Write the results using eponents. 6. ( )( ). z. ( ab) ( 6a b ). ( ) 0
53 Eercise Evaluate each epression ( 8) when 7. when and Simplif each epression. 8. ( )( 7 ) 9. (z )(z )(z ) ( 0 z ). z
54 . Negative Eponents and Scientific Notation Learning Objectives:. Simplif epressions containing negative eponents.. Use the rules and definitions for eponents to simplif eponential epressions.. Convert numbers in standard form to scientific notation.. Convert numbers in scientific notation to standard form.. Ke Vocabular: negative eponents, scientific notation, standard form. Properties of Negative Eponents: If m and n are integers, and and are an real number, 0, 0, then m n n.. n n.. m n m n m n mn m. ( ) 6. ( ) m m m m m m 7. m m m 8. n m n n m ( ) Eample. Write using positive eponents and simplif the following.... a. Eample. Write using negative eponents
55 Eample. Performed the indicated operation and simplif. Write answer using positive eponents a b 6 a b Scientific Notation is an epression of the form: n a 0 where a < 0 and n is the power.. Convert a number to scientific notation Steps:. If decimal point in the given number moves to left, n is positive.. If decimal point in the given number moves to right, n is negative. Eample. Write the given number in scientific notation.. 6,0, Convert scientific notation to a number Steps:. If n is positive, moves decimal point in the given number to the right.. If n is negative, move decimal point in the given number to left.
56 Eample. Write the given scientific notation in standard notation Eercise Simplif each epression. Write results with positive eponents p. 7 q 6. 9 Write each number in scientific notation.. ( ) 7. ( ) ( ) 9.,000,000, Write each number in standard form. 8. ( a b c) ( a b)
57 . Introduction to Polnomial Learning Objectives:. Define term and coefficient of a term.. Define polnomial, monomial, binomial, trinomial, and degree.. Evaluate polnomials for given replacement values.. Simplif a polnomial b combining like terms.. Simplif a polnomial in several variables. 6. Write a polnomial in descending powers of the variable and with no missing powers of the variable. 7. Ke Vocabular: coefficient, constant, polnomial, monomial, binomial, trinomial, degree of a term, degree of the polnomial. A. Classifing Polnomial n Polnomial is a finite sum of terms of the form a, where a is a real number and n is a whole number. Term is a number or the product of a number and variables raised to powers separated b plus or minus signs. Numerical Coefficient (coefficient) is the numerical factor of each term. Constant term is the term that contains onl a number. Tpes of Polnomials. Monomial is a polnomial with one term.. Binomial is a polnomial with two terms.. Trinomial is a polnomial with three terms.. Polnomial is a polnomial with four or more terms. Eample. Classif as monomial, binomial, or trinomial ( 9) ( ) B. Finding the Degree of a Polnomial Degree of a Polnomial is the greatest degree of an term of the polnomial. Note:. A constant term has zero degree.. Zero polnomial has no degree. Eample. Find the degree of the terms and the degree of the polnomial.. 9. z z 8. 7
58 C. Writing a Polnomial in Descending Order Eample. Write in descending order:. 8. D. Evaluating Polnomials Eample. Find the value of P( t) 6t 90, when t E. Simplifing Polnomials b Combining Like Terms Like Terms are terms that contain eactl the same variables raised to eactl the same powers. To Combine Like Terms is to combine the coefficient of the like term. Eample. Simplif b combining like terms
59 . 7 Eample. Find the total area of the rectangles. 6. Eercise Simplif each epression. Write results with positive eponents.. Complete the table for the polnomial 7 Term 7 Coefficient Find (a) the degree of each of each term (b) the degree of the following polnomials (c) determine whether it is a monomial, binomial, trinomial, or none of these a b 7 a b a 9
60 Find the value of each polnomial when (a) 0 and (b) Simplif each of the following b combining like terms
61 . Adding and Subtracting Polnomials Learning Objectives:. Add polnomials.. Subtract polnomials.. Add or subtract polnomials in one variable.. Add or subtract polnomials in several variables.. Ke Vocabular: combine like terms. A. Adding or subtracting Polnomials To Add or to subtract Polnomials is to add or subtract the coefficient of the like terms. Eample. Perform the indicated operation. Add: 8 and 8. Subtract 8 from 6. Subtract from the sum of and.. ( a ab 6b ) ( a ab 7b ). ( 9 ) ( 7 8 ) 6
62 6. Eercise Add or subtract as indicated.. ( ) ( ) 9 7. ( ) ( ) 9 7 a a a a. ( ) ( ) 7 a a a. ( ) ( ) ( ) a a a a a ( ) ( ) 7 7 7
63 . Multipling Polnomials Learning Objectives. Multipl monomials.. Multipl a monomial b a polnomial.. Multipl two polnomials.. Multipl polnomials verticall.. Ke Vocabular: polnomial, monomial, binomial, trinomial. A. Multipling Two Monomials Steps.. Multipl the coefficient with the coefficient.. Multipl the like variable b adding their power. Eample. Multipl:. ( )( ). ( )( ) B. Multipling a Monomial and a Binomial Distributive Propert: a ( b c) ab ac and a( b c) ab ac Eample. Multipl:. ( a b). ( ) C. Multipling Two Binomials Using Distributive Propert ( a b)( c d) a ( c d ) b( c d ) ac ad bc bd Eample. Multipl. 6
64 . ( a )( a ). ( ). ( )( ). Eercise (. )( ). ( ab b ) a. ab ( a 7a b 9b ) ( )( )( ) 6. ( 7 )( ) 8. ( )( ) 0. ( )( ). Find the area of a rectangle with sides ( ) ft. and ( 7). Find the area of a triangle with height 6 in. and base ( 9 ) ft. in. 6
65 .6 Special Products of Polnomials Learning Objectives:. Multipl two binomials using the FOIL Method.. Square a binomial.. Multipl the sum and difference of two terms.. Use special products to multipl binomials.. Ke Vocabular: FOIL method, squaring a binomial. Special Products of Polnomials:. Product of Two Binomials: ( a b)( c d ) ac ad bd bc. The Square of a Binomial Sum: ( a b) ( a b)( a b) a ab b. The Square of a Binomial Difference: ( a b) ( a b)( a b) a ab b. The Product of the Sum and Difference of Two Terms: ( )( ) a b a b a b Eample. Multipl.. ( 9)( ). ( ). ( ). 6
66 . ( )( ).6 Eercise Multipl.. ( 0). ( ). ( 7 )( 7 ) 6. ( )( 8 ) a a a. ( 0)( 0) 66
67 .7 Division of Polnomials Learning Objectives:. Divide a polnomial b a monomial.. Use long division to divide a polnomial b a polnomial other than a monomial.. Ke Vocabular: dividend, quotient, divisor. A. Dividing a Polnomial b a Monomial A B A B Addition Rule: ; C 0 C C C Eample. Divide the following b 9 B. Dividing a Polnomial b a Binomial Eample. Divide 6 b 67
68 .7 Eercise Divide p p 7 p. 6 6 a a a 0 a
69 Chapter Factoring Polnomials. The Greatest Common Factors Learning Objectives:. Find the greatest common factor of a list of numbers.. Find the greatest common factor of a list of terms. Factor out the greatest common factor from the terms of a polnomial.. Factor b grouping.. Ke Vocabular: factors, factored form, factoring, factoring out, greatest common factor, GCF, factoring b grouping. A. Finding the Greatest common Factor (GCF) of Numbers Greatest common Factor (GCF) is the largest common factor of the integers in the list. Steps to find the GCF.. Write each of the numbers as a product of prime number using eponent for repeated number.. Choose the number that has the lowest eponent, then find their product.. For the common variable, choose the smallest eponent Eample. Find the GCF of:., 60 and 08. 8, 9 and
70 B. Factoring Out the Greatest Common Factor Eample. Factor completel:. 6 a a a. ( ) ( ). Eercise Find the GCF for each list..,. 8 7 a, a, a., 7, 8 Factor out the GCF from each polnomial.. a ( ) ( ) Factor the following four term polnomials b grouping. 9. a ( b ) ( b )
71 . Factoring Trinomial of the Form b c Learning Objectives:. Factor trinomials of the form b c.. Factor out the greatest common factor and then factor a trinomial of the form b c.. Ke Vocabular: factor, product, sum, binomials, prime. A. Factoring Trinomials of the Form b c Steps for Factoring b c : Find two integers whose product is c and whose sum is b.. If b and c are positive, then both integers must be positive. b c ( )( ). If c is positive and b is negative, then both integers must be negative. b c ( )( ). If c is negative, one integer must be positive and one negative. b c bigger number is ; smaller number is b c Eample. Factor completel:. 8 7 ( )( ) ( )( ) smaller number is ; bigger number is b b 0b 7
72 . a 6 8a. Eercise Factor the following trinomials completel. Write prime if the do not factor
73 . FACTORING TRINOMIALS OF THE FORM a b c, a Learning Objectives. Factor trinomials of the form a b c, a.. Factor out the GCF before factoring a trinomial of the form a b c.. Ke Vocabular: greatest common factor, coefficient. A. The ac Test for a b c ac TEST for Factoring of a b c a b c is factorable if there is two integers with product ac and sum is b. Eample. Determine whether the polnomial is factorable:... 9 B. Factoring a b c b trial and error Eample. Factor completel:
74 Eercise Complete the following.. 6 ( )( ). 8 ( )( ). 0 ( )( ). ( )( ) Factor the following trinomials completel. Write prime if the do not factor a a 0.. a a. a a. a a a 6a a
75 . Factoring Trinomials of the Form a b c, a b Grouping Learning Objectives. Use the grouping method to factor trinomials of the form a b c.. Ke Vocabular: grouping method, trinomial. A. Factor using the Grouping Method Step for factoring a trinomial b grouping. Factor out a greatest common factor, if there is one other than.. For the resulting trinomial a b c, a, find two numbers whose product is ac and whose sum is b.. Write the middle term, b, using the factors found in step.. Factor b grouping. Eample. Factor completel
76 . 0. Eercise Factor b grouping n 7n 7. m 7m a 0a 6 9. a a 6a 0. 60m m 6m
77 . Factoring Squares of Binomials Learning Objectives:. Recognize perfect square trinomials.. Factor perfect square trinomials.. Factor the difference of two squares.. Ke Vocabular: perfect squares, perfect square trinomial, square of a binomial, difference of two squares. Factoring Perfect Square Trinomials. Perfect Square Trinomials. a ab b a b a b a b a. Difference of Two Squares. a b a b a b ( )( ) ( ) ( a b)( a b) ( a ) ab b b ( )( ) a b is prime (cannot be factored) To be a perfect square trinomial, a trinomial must satisf conditions:. The first and last terms ( a and b ) must be perfect squares. There must be no minus signs before a and b. The middle term is ab or ab Eample. Determine whether the epression is the perfect square: Eample. Factor completel:
78
79 . Eercise. Is 0 a perfect square trinomial? Fill in the blank so that the trinomial is a perfect square trinomial Factor completel a a 7a a 60a 0a
80 .6 Solving Quadratic Equations b Factoring Learning Objectives:. Solve quadratic equations b factoring.. Solve equations with degree greater than b factoring.. Ke Vocabular: quadratic equation, standard form, zero factor propert, GCF. Quadratic Equation in Standard Form is an equation of the form a b c 0, where a, b, c are real numbers. Principle of Zero Product: If ab 0 then a 0 or b 0 Steps for Solving Quadratic Equations. Perform the necessar operations on both sides of the equation so that the right hand side is 0.. Use the general factoring strateg to factor the left side of the equation, if necessar.. Use the Principle of Zero Product to solve each of the resulting equations.. Check the results b substituting the solutions obtained in step in the original equation. Eample. Solve each of the following. ( 9 )( 0 )
81 . ( ). ( )( ) ( ) 7.6 Eercise Solve each equation.. ( )( ) 0. ( )( ) 0. ( )( ) ( 7)( ) m m n 8n n p p. w 0w 80w 8
82 .7 Quadratic Equations and Problem Solving Learning Objectives:. Translate words to algebraic epressions.. Solve problems that can be modeled b quadratic equations.. Ke Vocabular: Pthagorean Theorem, right triangle, hpotenuse, legs. Consecutive Numbers. Consecutive Number. Consecutive Odd Number. Consecutive Even Number st number st number st number nd number nd number nd number rd number rd number rd number th number th number 6 th number 6 Eample. The square of a number minus twice the number is 6. Find the numbers. Eample. The length of a rectangular garden is feet more than its width. The area of the garden is 76 square feet. Find the length and the width. Eample. Find two consecutive odd integers whose product is more that their sum. 8
83 Eample. A student dropped a ball from the top of a 6 foot building. The height of the ball after t seconds is given b the quadratic equation h 6t 6. How long will it take the ball to hit the ground? Pthagorean Theorem In an right triangle Leg a Hpotenuse c a b c Leg b Eample. The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hpotenuse is m. Find the lengths of the legs. 8
84 .7 Eercise Solve.. A rectangle has an area of square inches. The width is represented b and the length is. Find the dimensions.. The length of a rectangle is cm more than the width. The area is 70 cm. Find the dimensions of the rectangle.. The length of a proposed rectangular flower garden is 6 m more that its width. The area of the proposed garden is 7m. Find the dimensions of the proposed flower garden.. A square field had m added to its length and m added to its width. The field then had an area of 0 m. Find the length of a side of the original field.. A rock is dropped from a 78 foot cliff. The height h of the rock after t seconds is given b the equation h 6t 78. Find out how man seconds pass before the rock reaches the ground. 6. One leg of a right triangle measures 6 m while the length of the other leg measures meters. The hpotenuse measures ( 6) m. Find the length of all sides. 7. The longer leg of a right triangle measures two feet more than twice the length of the shorter leg. The hpotenuse measures feet more than twice the shorter leg. Find the length of all three sides. 8. Find the length of a ladder leaning against a building if the top of the ladder touches the building at a height of feet. Also, the length of the ladder is feet more than its distance from the base of the building. 9. One leg of a right triangle is meters longer than the other leg. The hpotenuse is 6 meters long. Find the length of each leg. 0. Eight more than the square of a number is the same as si times the number. Find the number.. Fifteen less than the square of a number is the same as twice the number. Find the numbers.. Seven less than times the square of a number is 8. Find the number.. Find two consecutive positive odd integers whose product is.. The sum of the squares of two consecutive integers is. Find the integers.. Find two consecutive odd integers such that the square of the first added to times the second is. 8
85 Chapter 6 Rational Epressions and Functions 6. Simplifing Rational Epressions Learning Objectives:. Find the value of a rational epression given a replacement number.. Identif values for which a rational epression is undefined.. Simplif or write rational epressions in lowest terms.. Write equivalent forms of rational epressions.. Ke Vocabular: rational epressions, simplifing rational epressions. P Rational Epression is an epression of the form ; P and Q are an polnomials; Q 0 Q A. Evaluating Rational Epressions Standard Form of a Fraction: For b 0. a a a a. b b b b a b a b a b a b Eample. Find the value of ; B. Identifing When a Rational Epression is Undefined A rational epression is undefined when the denominator is 0. Eample. Find values for which the rational epression is undefined.. m m 8
86 .. n n 9 C. Simplifing Rational Epressions. Fundamental Rule of Fractions: AC A BC B B 0, C 0. Quotient of Additive Inverses ( trick ) a b b a a. b a a b a b b a b. b a a b Step to Simplif a Rational Epression. Completel factor the numerator and denominator.. Cancel the common factor in the numerator and denominator b replace the quotient of the a common factors b the number, since. a. Rewrite the epression in simplified form. Eample. Simplif.. 6 6( m n ). ( m n) 86
87
88 6. Eercise Find the value of the following rational epressions when and Find an real number for which each rational epression is undefined Simplif each rational epression a 8b a b
Glossary. Also available at BigIdeasMath.com: multilanguage glossary vocabulary flash cards. An equation that contains an absolute value expression
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationmath0320 FALL interactmath sections developmental mathematics sullivan 1e
Eam final eam review 180 plus 234 TSI questions for intermediate algebra m032000 013014 NEW Name www.alvarezmathhelp.com math0320 FALL 201 1400 interactmath sections developmental mathematics sullivan
More informationreview math0410 (1174) and math 0320 ( ) aafinm mg
Eam Name review math04 (1174) and math 0320 (17243) 03201700aafinm0424300 mg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif. 1) 7 23 A)
More informationreview for math TSI 182 practice aafm m
Eam TSI 182 Name review for math TSI 182 practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Simplif.
More informationReview of Elementary Algebra Content
Review of Elementar Algebra Content 0 1 Table of Contents Fractions...1 Integers...5 Order of Operations...9 Eponents...11 Polnomials...18 Factoring... Solving Linear Equations...1 Solving Linear Inequalities...
More informationAlgebra II Notes Unit Five: Quadratic Functions. Syllabus Objectives: 5.1 The student will graph quadratic functions with and without technology.
Sllabus Objectives:.1 The student will graph quadratic functions with and without technolog. Quadratic Function: a function that can be written in the form are real numbers Parabola: the Ushaped graph
More informationEam Name algebra final eam review147 aam032020181t4highschool www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation.
More informationAlgebra II Notes Unit Six: Polynomials Syllabus Objectives: 6.2 The student will simplify polynomial expressions.
Algebra II Notes Unit Si: Polnomials Sllabus Objectives: 6. The student will simplif polnomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a and
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1 Final Eam Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve. 1)   6 =  + 7 1) ) 6 + 7(  ) = 8  ) )  t + t = 6 t + 1 ) Solve
More informationMath Intermediate Algebra
Math 095  Intermediate Algebra Final Eam Review Objective 1: Determine whether a relation is a function. Given a graphical, tabular, or algebraic representation for a function, evaluate the function and
More informationSolve each system by graphing. Check your solution. y =3x x + y = 5 y =7
Practice Solving Sstems b Graphing Solve each sstem b graphing. Check our solution. 1. = + 3 =  (1, ). = 1  (, 1) =3 + 5 3. = 3 + + = 1 (, 3). =5 =  7. = 35 3  = 0 (1, 5) 5. 3 + = 5 =7 (, 7).
More informationMath 0308 Final Exam Review(answers) Solve the given equations. 1. 3x 14 8x 1
Math 8 Final Eam Review(answers) Solve the given equations.. 8.. 9.. 9 9 8 8.. 8 8 all real numbers 8. 9. all real numbers no solution 8 8 9 9 9 Solve the following inequalities. Graph our solution on
More informationMini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models
Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution
More informationC) x m A) 260 sq. m B) 26 sq. m C) 40 sq. m D) 364 sq. m. 7) x x  (6x + 24) = 4 A) 0 B) all real numbers C) 4 D) no solution
Sample Departmental Final  Math 46 Perform the indicated operation. Simplif if possible. 1) 7   22 + 3 2  A) +  2 B)  + 42 C) + 42 D)  +  2 Solve the problem. 2) The sum of a number and its
More informationmath FALL developmental mathematics sullivan 1e
TSIpractice eam review 1 131 180 plus 34 TSI questions for elementar and intermediate algebra m0300004301 aaa Name www.alvarezmathhelp.com math0300004301 FALL 01 100 interactmath developmental mathematics
More informationreview for math TSI 55 practice aafm m
Eam TSI Name review for math TSI practice 01704041700aafm042430m www.alvarezmathhelp.com MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the
More informationDiagnostic Tests Study Guide
California State Universit, Sacramento Department of Mathematics and Statistics Diagnostic Tests Stud Guide Descriptions Stud Guides Sample Tests & Answers Table of Contents: Introduction Elementar Algebra
More informationNOTES. [Type the document subtitle] Math 0310
NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) C) 31.
Eam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Write the sentence as a mathematical statement. 1) Negative twentfour is equal to negative
More informationMini Lecture 1.1 Introduction to Algebra: Variables and Mathematical Models
Mini Lecture. Introduction to Algebra: Variables and Mathematical Models. Evaluate algebraic epressions.. Translate English phrases into algebraic epressions.. Determine whether a number is a solution
More informationMath 100 Final Exam Review
Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.
More informationCourse 15 Numbers and Their Properties
Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks.
More informationLESSON #28  POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #8  POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationStudy Guide for Math 095
Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.
More informationAlgebra 2 Semester Exam Review
Algebra Semester Eam Review 7 Graph the numbers,,,, and 0 on a number line Identif the propert shown rs rs r when r and s Evaluate What is the value of k k when k? Simplif the epression 7 7 Solve the equation
More informationBRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE. MTH 05 Review Sheet
BRONX COMMUNITY COLLEGE of the Cit Universit of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH 05 Review Sheet Go to http://www.cun.edu/testing for more information on the CUNY Elementar Algebra
More informationMiniLecture 8.1 Solving Quadratic Equations by Completing the Square
MiniLecture 8.1 Solving Quadratic Equations b Completing the Square Learning Objectives: 1. Use the square root propert to solve quadratic equations.. Solve quadratic equations b completing the square.
More information2.1 Evaluate and Graph Polynomial
2. Evaluate and Graph Polnomial Functions Georgia Performance Standard(s) MM3Ab, MM3Ac, MM3Ad Your Notes Goal p Evaluate and graph polnomial functions. VOCABULARY Polnomial Polnomial function Degree of
More informationWoodland Community College: Math Practice Test
Woodland Communit College: Math Practice Test Elementar Algebra Math Test The following problems are recommended practice problems for the elementar algebra section of the placement test. Some of the problems
More informationOn a video game, Jacob got 1685 points and earned two bonuses worth 193 and 270 points. What is his total score? Answer: 2148 points
Chapter Numerical Expressions and Factors Information Frame 9. Sample answers are given.. Ke Words: the sum of, the total of RealLife Application : On a video game, Jacob got 68 points and earned two
More informationOBJECTIVES UNIT 1. Lesson 1.0
OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint
More informationMt. Douglas Secondary
Foundations of Math 11 Section.1 Review: Graphing a Linear Equation 57.1 Review: Graphing a Linear Equation A linear equation means the equation of a straight line, and can be written in one of two forms.
More informationMATH 080 FinalExam Review
MATH 080 FinalExam Review Can you simplify an expression using the order of operations? 1) Simplify 32(118)  18 3 23 2) Simplify 53 33 6 + 3 A) 5 9 B) 19 9 C)  25 9 D) 25 9 Can you evaluate an algebraic
More informationNorthwest High School s Algebra 2/Honors Algebra 2
Northwest High School s Algebra /Honors Algebra Summer Review Packet 0 DUE Frida, September, 0 Student Name This packet has been designed to help ou review various mathematical topics that will be necessar
More informationIntermediate Algebra 100A Final Exam Review Fall 2007
1 Basic Concepts 1. Sets and Other Basic Concepts Words/Concepts to Know: roster form, set builder notation, union, intersection, real numbers, natural numbers, whole numbers, integers, rational numbers,
More informationMATH 1710 College Algebra Final Exam Review
MATH 7 College Algebra Final Eam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) There were 80 people at a pla. The admission price was $
More informationMy Math Plan Assessment #1 Study Guide
My Math Plan Assessment #1 Study Guide 1. Find the xintercept and the yintercept of the linear equation. 8x y = 4. Use factoring to solve the quadratic equation. x + 9x + 1 = 17. Find the difference.
More informationReady To Go On? Skills Intervention 61 Polynomials
6A Read To Go On? Skills Intervention 6 Polnomials Find these vocabular words in Lesson 6 and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading
More informationLESSON #24  POWER FUNCTIONS COMMON CORE ALGEBRA II
1 LESSON #4  POWER FUNCTIONS COMMON CORE ALGEBRA II Before we start to analze polnomials of degree higher than two (quadratics), we first will look at ver simple functions known as power functions. The
More informationMAT Intermediate Algebra  Final Exam Review Textbook: Beginning & Intermediate Algebra, 5th Ed., by MartinGay
MAT0  Intermediate Algebra  Final Eam Review Tetbook: Beginning & Intermediate Algebra, 5th Ed., by MartinGay Section 2. Solve the equation. ) 9  (  ) = 2 Section 2.8 Solve the inequality. Graph the
More information2. Which of the following expressions represents the product of four less than three times x and two more than x?
Algebra Topics COMPASS Review You will be allowed to use a calculator on the COMPASS test. Acceptable calculators are: basic calculators, scientific calculators, and graphing calculators up through the
More informationDiaz Math 080 Midterm Review: Modules AF Page 1 of 7
Diaz Math 080 Midterm Review: Modules AF Page 1 of 7 1. Use the rule for order of operations to simplif the epression: 11 9 7. Perform the indicated operations and simplif: 7( + ) 6(5 9) 3. If a = 3,
More informationMath 030 Review for Final Exam Revised Fall 2010 RH/ DM 1
Math 00 Review for Final Eam Revised Fall 010 RH/ DM 1 1. Solve the equations: (1) (7) () (1) () 1 1 1 1 f. 1 g. h. 1 11 i. 9. Solve the following equations for the given variable: 1 Solve for. D ab
More informationMath 100 Final Exam Review
Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.
More informationAlgebra 2 Honors Summer Packet 2018
Algebra Honors Summer Packet 018 Solving Linear Equations with Fractional Coefficients For these problems, ou should be able to: A) determine the LCD when given two or more fractions B) solve a linear
More informationIntermediate Algebra Review for Exam 1  Spring 2005
Intermediate Algebra Review for Eam  Spring 00 Use mathematical smbols to translate the phrase. ) a) 9 more than half of some number b) 0 less than a number c) 37 percent of some number Evaluate the epression.
More informationRational Equations. You can use a rational function to model the intensity of sound.
UNIT Rational Equations You can use a rational function to model the intensit of sound. Copright 009, K Inc. All rights reserved. This material ma not be reproduced in whole or in part, including illustrations,
More informationAlgebra I Final Exam Review 2016 List of Topics Covered
Algebra I Final Exam Review 016 List of Topics Covered Know all vocabular, pa attention to the highlighted words in the text, and understand the various tpes of directions in each of the sections of the
More informationMath 154A Elementary Algebra Fall 2014 Final Exam Study Guide
Math A Elementar Algebra Fall 0 Final Eam Stud Guide The eam is on Tuesda, December 6 th from 6:00pm 8:0pm. You are allowed a scientific calculator and a " b 6" inde card for notes. On our inde card be
More informationREVIEW PACKET FOR END OF COURSE EXAM
Math H REVIEW PACKET FOR END OF COURSE EXAM DO NOT WRITE ON PACKET! Do on binder paper, show support work. On this packet leave all fractional answers in improper fractional form (ecept where appropriate
More informationIntroduction  Algebra I
LIFORNI STNRS TEST lgebra I Introduction  lgebra I The following released test questions are taken from the lgebra I Standards Test. This test is one of the alifornia Standards Tests administered as part
More information3.1 Graph Quadratic Functions
3. Graph Quadratic Functions in Standard Form Georgia Performance Standard(s) MMA3b, MMA3c Goal p Use intervals of increase and decrease to understand average rates of change of quadratic functions. Your
More informationOne of your primary goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan.
PROBLEM SOLVING One of our primar goals in mathematics should be to become a good problem solver. It helps to approach a problem with a plan. Step Step Step Step Understand the problem. Read the problem
More information118 Intermediate Algebra Fall Intermediate Algebra FNMT Time: 6:309:35 pm, Thursday. Room 228
118 Intermediate Algebra Fall 2018 Intermediate Algebra  40428  FNMT 118122 Time: 6:309:3 pm, Thursda Room 228 SYLLABUS Catalog description. Real numbers, polnomials, rational epressions, algebraic
More informationSummer Math Packet (revised 2017)
Summer Math Packet (revised 07) In preparation for Honors Math III, we have prepared a packet of concepts that students should know how to do as these concepts have been taught in previous math classes.
More informationPREALGEBRA SUMMARY WHOLE NUMBERS
PREALGEBRA SUMMARY WHOLE NUMBERS Introduction to Whole Numbers and Place Value Digits Digits are the basic symbols of the system 0,,,, 4,, 6, 7, 8, and 9 are digits Place Value The value of a digit in
More informationWest Campus State Math Competency Test Info and Practice
West Campus State Math Competenc Test Info and Practice Question Page Skill A Simplif using order of operations (No grouping/no eponents) A Simplif using order of operations (With grouping and eponents)
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 03 CFair College Departmental Final Eamination Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Graph the linear equation b the method of
More informationIntermediate Algebra Math 097. Evaluates/Practice Tests. For solutions, refer to the back of the PAN.
Intermediate Algebra Math 097 Evaluates/Practice Tests For solutions, refer to the back of the PAN. Page of 8 Take this practice test to be sure that ou are prepared for the final quiz in Evaluate.. Solve
More informationAlgebra 2 CPA Summer Assignment 2018
Algebra CPA Summer Assignment 018 This assignment is designed for ou to practice topics learned in Algebra 1 that will be relevant in the Algebra CPA curriculum. This review is especiall important as ou
More informationGlossary. Also available at BigIdeasMath.com: multilanguage glossary vocabulary flash cards
Glossar This student friendl glossar is designed to be a reference for ke vocabular, properties, and mathematical terms. Several of the entries include a short eample to aid our understanding of important
More informationLearning Goals. College of Charleston Department of Mathematics Math 101: College Algebra Final Exam Review Problems 1
College of Charleston Department of Mathematics Math 0: College Algebra Final Eam Review Problems Learning Goals (AL) Arithmetic of Real and Comple Numbers: I can classif numbers as natural, integer,
More informationMATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline
MATH 0960 ELEMENTARY ALGEBRA FOR COLLEGE STUDENTS (8 TH EDITION) BY ANGEL & RUNDE Course Outline 1. Real Numbers (33 topics) 1.3 Fractions (pg. 27: 175 odd) A. Simplify fractions. B. Change mixed numbers
More informationName Period Date. Practice FINAL EXAM Intro to Calculus (50 points) Show all work on separate sheet of paper for full credit!
Name Period Date Practice FINAL EXAM Intro to Calculus (0 points) Show all work on separate sheet of paper for full credit! ) Evaluate the algebraic epression for the given value or values of the variable(s).
More informationChapter 12 Add and Subtract Integers
Chapter 12 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign
More informationReady To Go On? Skills Intervention 51 Using Transformations to Graph Quadratic Functions
Read To Go On? Skills Intervention 51 Using Transformations to Graph Quadratic Functions Find these vocabular words in Lesson 51 and the Multilingual Glossar. Vocabular quadratic function parabola verte
More informationEvaluate algebraic expressions for given values of the variables.
Algebra I Unit Lesson Title Lesson Objectives 1 FOUNDATIONS OF ALGEBRA Variables and Expressions Exponents and Order of Operations Identify a variable expression and its components: variable, coefficient,
More information1) (3) + (6) = 2) (2) + (5) = 3) (7) + (1) = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = 7) (5)(4) = 8) (3)(6) = 9) (1)(2) =
Extra Practice for Lesson Add or subtract. ) (3) + (6) = 2) (2) + (5) = 3) (7) + () = 4) (3)  (6) = 5) (+2)  (+5) = 6) (7)  (4) = Multiply. 7) (5)(4) = 8) (3)(6) = 9) ()(2) = Division is
More informationPrealgebra and Elementary Algebra
Prealgebra and Elementary Algebra 9781635450897 To learn more about all our offerings Visit Knewton.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Lynn Marecek, Santa
More informationMATH 110: FINAL EXAM REVIEW
MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set
More informationAn assessment of these skills will occur the first week of school.
Fort Zumwalt West High School PreAP Algebra SUMMER REVIEW PACKET Name: * This packet is to be handed in to our Pre AP Algebra teacher on the first da of the school ear. *All work must be shown in the
More informationEquations and Inequalities
Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types
More informationReady To Go On? Skills Intervention 21 Solving Linear Equations and Inequalities
A Read To Go n? Skills Intervention 1 Solving Linear Equations and Inequalities Find these vocabular words in Lesson 1 and the Multilingual Glossar. Vocabular equation solution of an equation linear
More information1. Simplify each expression and write all answers without negative exponents. for variable L.
MATH 0: PRACTICE FINAL Spring, 007 Chapter # :. Simplif each epression and write all answers without negative eponents. ( ab ) Ans. b 9 7a 6 Ans.. Solve each equation. 5( ) = 5 5 Ans. man solutions + 7
More informationMath 0200 Final Exam Review Questions
Math 000 Final Eam Review Questions 1. Simplif: 4 8i + 8 ( 7). Simplif: 11 ( 9) + 6(10 4) + 4. Simplif ( 5 + 7) ( ) 8 6 4. Simplif: (4 ) 9 i 5. Simplif: 4 7 6. Evaluate 4 + 5 when = and = Write each of
More informationAP Calculus AB Summer Assignment Mrs. Berkson
AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre Calculus skills required for AP Calculus AB. Net ear we will be starting off the
More informationMath 75 MiniMod Due Dates Spring 2016
MiniMod 1 Whole Numbers Due: 4/3 1.1 Whole Numbers 1.2 Rounding 1.3 Adding Whole Numbers; Estimation 1.4 Subtracting Whole Numbers 1.5 Basic Problem Solving 1.6 Multiplying Whole Numbers 1.7 Dividing
More informationChapter 8 Vocabulary Check
28 CHAPTER 8 Quadratic Equations and Functions d. What is the level of methane emissions for that ear? (Use our rounded answer from part (c).) (Round this answer to 2 decimals places.) Use a graphing calculator
More informationDiaz Math 080 Midterm Review: Modules AF Page 1 of 7
Diaz Math 080 Midterm Review: Modules AF Page 1 of 7 1. Use the rule for order of operations to simplif the epression: 11 9 7. Perform the indicated operations and simplif: 7(4 + 4) 6(5 9) 3. If a = 3,
More informationMATH 91 Final Study Package Name
MATH 91 Final Stud Package Name Solve the sstem b the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to epress the solution set. 1)  = 1 1)
More informationMath 0240 Final Exam Review Questions 11 ( 9) 6(10 4)
Math 040 Final Eam Review Questions 11 ( 9) 6(10 4) 1. Simplif: 4 8 3 + 8 ( 7). Simplif: 34 3. Simplif ( 5 7) 3( ) 8 6 4. Simplif: (4 3 ) 9 5. Simplif: 6. Evaluate 4 7 3 3 4 5 when and 3 Write each of
More informationCHAPTER 2 Polynomial and Rational Functions
CHAPTER Polnomial and Rational Functions Section. Quadratic Functions..................... 9 Section. Polnomial Functions of Higher Degree.......... Section. Real Zeros of Polnomial Functions............
More informationBishop Kelley High School Summer Math Program Course: Algebra 2 A
06 07 Bishop Kelley High School Summer Math Program Course: Algebra A NAME: DIRECTIONS: Show all work in packet!!! The first 6 pages of this packet provide eamples as to how to work some of the problems
More informationState whether the following statements are true or false: 27.
Cumulative MTE 9 Review This packet includes major developmental math concepts that students ma use to prepare for the VPT Math (Virginia Placement Test for Math or for students to use to review essential
More informationElementary Algebra
Elementary Algebra 9781635450682 To learn more about all our offerings Visit Knewton.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Lynn Marecek, Santa Ana College MaryAnne
More informationAdvanced Algebra Scope and Sequence First Semester. Second Semester
Last update: April 03 Advanced Algebra Scope and Sequence 034 First Semester Unit Name Unit : Review of Basic Concepts and Polynomials Unit : Rational and Radical Epressions Sections in Book 0308 SLOs
More information= 9 = x + 8 = = 5x 19. For today: 2.5 (Review) and. 4.4a (also review) Objectives:
Math 65 / Notes & Practice #1 / 20 points / Due. / Name: Home Work Practice: Simplify the following expressions by reducing the fractions: 16 = 4 = 8xy =? = 9 40 32 38x 64 16 Solve the following equations
More informationDirection: Please write your answer in the answer blanks and show all work to get full credits. Each question is worth 4 points each.
EAST LOS ANGELES COLLEGE NAME: MATH 110 (INTRO TO MATH CONCEPTS) FINAL EXAM SAMPLE INSTRUCTOR: ANNE SISWANTO; TOTAL POINTS: 100; TIME: 10 MINUTES. Direction: Please write our answer in the answer blanks
More informationGlossary. Glossary 981. Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle
More informationAP Calculus AB Summer Assignment Mrs. Berkson
AP Calculus AB Summer Assignment Mrs. Berkson The purpose of the summer assignment is to prepare ou with the necessar Pre Calculus skills required for AP Calculus AB. Net ear we will be starting off the
More informationWe are working with degree two or
page 4 4 We are working with degree two or quadratic epressions (a + b + c) and equations (a + b + c = 0). We see techniques such as multiplying and factoring epressions and solving equations using factoring
More informationx. 4. 2x 10 4x. 10 x
CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC91.A.REI.1 Eplain
More informationMATH 103 Sample Final Exam Review
MATH 0 Sample Final Eam Review This review is a collection of sample questions used b instructors of this course at Missouri State Universit. It contains a sampling of problems representing the material
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C)b = h 2A
MATH 830/GRACEY EXAM PRACTICE/CH..3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the formula for the specified variable. 1) A = 1 bh
More informationAnswer Explanations. The SAT Subject Tests. Mathematics Level 1 & 2 TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE
The SAT Subject Tests Answer Eplanations TO PRACTICE QUESTIONS FROM THE SAT SUBJECT TESTS STUDENT GUIDE Mathematics Level & Visit sat.org/stpractice to get more practice and stud tips for the Subject Test
More informationCHAPTER 2 Solving Equations and Inequalities
CHAPTER Solving Equations and Inequalities Section. Linear Equations and Problem Solving........... 8 Section. Solving Equations Graphically............... 89 Section. Comple Numbers......................
More informationState whether the following statements are true or false: 30. 1
Cumulative MTE 9 Review This packet includes major developmental math concepts that students ma use to prepare for the VPT Math (Virginia Placement Test for Math or for students to use to review essential
More informationMA094 Part 2  Beginning Algebra Summary
MA094 Part  Beginning Algebra Summary Page of 8/8/0 Big Picture Algebra is Solving Equations with Variables* Variable Variables Linear Equations x 0 MA090 Solution: Point 0 Linear Inequalities x < 0 page
More informationChapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs
Ch 5 Alg Note Sheet Ke Chapter 5: Quadratic Equations and Functions 5.1 Modeling Data With Quadratic Functions Quadratic Functions and Their Graphs Definition: Standard Form of a Quadratic Function The
More informationPart 2  Beginning Algebra Summary
Part  Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More information