California State Universit, Northridge MATH 09 HYBRID WORKBOOKS Spring 00
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.
Eample. Solve each equation.. 8 6 0..9 a.7 a. 7a.. 6 6. ( ) ( )
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.. 8. 9. r.. 7. k 8. 6. 9 7. 8 8. p 7 0p 9 9. 9
0. 6 7. 7.7.8. 7. 6. 9 ( 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.
. 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....8.9... ( ) 8 ( ) 6
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. 9. 8 8. 7. 8. d 7. 7 0. 7 p. 0 7 6.. 0. 7 8. 6 9. 9 8. 9 0.7k 9. b.. 7 z 8 z 0. 6 6. 0. 0. 0 7. 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
. 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
. 0.0 0.06( 00) 70. 6. Eercise Solve each equation.. 7 8. ( 7) 0. ( n ) ( 7n ) 8. 6( ) ( ). 7 ( ) ( 8) 6. ( ) 7 7. 8 6 8. ( 9 a ) a a 9. 0. 0. 6. 8 9 9 k k 6 ( ) ( ). 0.76 0.( 80) 0.( 68). 0.( 00) 0.0 0. 0.07( 00). 0 ( ) 7 6. 6 7. 8 ( 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
. 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
. 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.
. 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 9. 9. 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.
. 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.
. 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.)
. 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. 6. 7. 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. 9 0. 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
.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. -6 - - 0 6. m -6 - - 0 6. < t 0-6 - - 0 6 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
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 -6 - - 0 6. 9-0 -0-0 0 0 0 0. 8 ( ) -6 - - 0 6 8 7. ( ) > ( ) -6 - - 0 6 7
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. 7 < 6 6. < 7. 8. 0.7 >. 9. 7 8
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
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. 0 0 0
. 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.. 8. 0 0 0
. 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...
.. Eercise For each equation, find three ordered pair solutions b completing the table. Then, use the ordered pairs to graph the equation.. 7. 0 0 0. 8 0 Graph each linear equation. Label at least three points on the graph grid... 6 6. 0 7. 8. 6 9. 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.
. 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.. 0. 0
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.. 0. 0. Eercise Identif the intercepts and intercepts points..
... 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.. 6. 6 7. 6 8 8. 9. 0. 6. 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
. 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
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
. 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
Find the slope of each line. 8. 9. 7 0. 0. 8 Determine whether the lines are parallel, perpendicular, or neither... 8. 7 9 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
. 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.
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.. 6. 7. 8. 8 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 ( )
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.
.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),(, ) }. (, ) {, }. ( )
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...
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.. 6. 6
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
.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... 0 8
. < 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.. >. < 6. 7. 8 6 9
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
. 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.. 0. 6 8
. 8. Eercise Determine whether the ordered pair satisfies the sstem of linear equations.. ( ),. 0 6 ( ), 0. 6 9 ( ) 6, Solve each sstem of equations b graphing. State the solution(s) and tpe of sstem... 6 6. 7. 8 8. 9.
. 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.. 6 6. 6
. Eercise Solve each sstem of equations b substitution. State the solution(s) and tpe of sstem... 7. 6 6. 8. 6. 6 7 7. 7 7 8. 6 7 0.
. 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.. 6. 0. 0. 0.6 0.0 0.0 0.
6. 6. Eercise Solve each sstem of equations b addition. State the solution(s) and tpe of sstem.. 9. 0. 8 6. 6. 8 8 6 b a b a 6. 0 6 8 7 7. 6 8. 9. 8
. 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
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
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
. 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
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 6. 7. 0 8. ( ) 0 m m m m m CAUTION! ( ) and ( ) m Eample. Evaluate each epression.. ( 7). 6.
.. 8 8 0 6. when Eample. Use the properties of the eponent to simplif. Write the results using eponents. 6. ( )( ). z. ( ab) ( 6a b ). ( ) 0
. 0 9 0. Eercise Evaluate each epression.. 8. 8. ( 8). 7. 6. 6 when 7. when and Simplif each epression. 8. ( )( 7 ) 9. (z )(z )(z ) 0.. 0. ( 0 z ). z 7 0 7 8.
. 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 0 9. 0 0. 0. ( ) Eample. Write using positive eponents and simplif the following.... a. Eample. Write using negative eponents.. 8 7. 6
Eample. Performed the indicated operation and simplif. Write answer using positive eponents.. 6. 7 7. 6. 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.. 6,0,000. 0.00. 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.
Eample. Write the given scientific notation in standard notation.. 6.6 0.. 0. Eercise Simplif each epression. Write results with positive eponents... 6. p. 7 q 6. 9 Write each number in scientific notation.. ( ) 7. ( ) ( ) 9.,000,000,000 0. 0.000 Write each number in standard form. 8. ( a b c) ( a b)..78 7 0. 6.0 0 0 6
. 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.. 6. 8( 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
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.. 6..7 6..6.8 8.. 7 8 8
. 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.. 7.. a b 7 a b a 9
Find the value of each polnomial when (a) 0 and (b). 6. 6 7. 7 Simplif each of the following b combining like terms. 8. 9 9. 7 7 6 0...6.. 60
. 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
6. Eercise Add or subtract as indicated.. ( ) ( ) 9 7. ( ) ( ) 9 7 a a a a. ( ) ( ) 7 a a a. ( ) ( ) ( ) 8 6 7 a a a a a. 6 9 8 6. ( ) ( ) 7 7 7
. 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
. ( a )( a ). ( ). ( )( ). Eercise. 6 9 0. (. )( ). ( ab b ) a. ab ( a 7a b 9b ) 7. 7. ( )( )( ) 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
.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
. ( )( ).6 Eercise Multipl.. ( 0). ( ). ( 7 )( 7 ) 6. ( )( 8 ) a. 7 8 7. a a. ( 0)( 0) 66
.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. 8.. 7 9 b 9 B. Dividing a Polnomial b a Binomial Eample. Divide 6 b 67
.7 Eercise Divide.. 6 8 6. p p 7 p. 6 6 a. 7. 7 6. 0a a 0 a 7. 6 0 68
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 8 0 69
B. Factoring Out the Greatest Common Factor Eample. Factor completel:. 6 a 8. 6 6. 7 6 0 0. a a. ( ) ( ). Eercise Find the GCF for each list..,. 8 7 a, a, a., 7, 8 Factor out the GCF from each polnomial.. a. 6 7 6. 6 7. 7 8. ( ) ( ) Factor the following four term polnomials b grouping. 9. a ( b ) ( b ) 0.. 7. 6 0. 0 70
. 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. 8 9.. 6 b b 0b 7
. a 6 8a. Eercise Factor the following trinomials completel. Write prime if the do not factor.. 6. 6. 6 7. 08. 7 6. 0 7. 8. 6 9. 60 0.. 6. 0 0. 6 0.. 9 7
. 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:.. 8 6 7
. 6 6.. Eercise Complete the following.. 6 ( )( ). 8 ( )( ). 0 ( )( ). ( )( ) Factor the following trinomials completel. Write prime if the do not factor.. 7 6. 8 0 7. 6 0 8. 9 8 8 9. 7 0. 8a a 0.. a a. a a. a a. 9 6. 7. 8 0 8. 0a 6a a 9. 0 6 0 7
. 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.. 9. 0 0. 6. 7 9 6 7
. 0. Eercise Factor b grouping.... 6 8. 7. 9 6. 0n 7n 7. m 7m 8 8. 6a 0a 6 9. a a 6a 0. 60m m 6m. 8 98. 0 6. 6 76
. 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:. 9. 9 9 Eample. Factor completel:. 6 9 77
. 9. 9 0. 9. 9 6. 8 7. 78
. Eercise. Is 0 a perfect square trinomial? Fill in the blank so that the trinomial is a perfect square trinomial. 6. 9 Factor completel.. 9. 9 0. 6 9. 0 60. a a 7a 6. 7. 8. 9 9 9. 8a 60a 0a 6 0.. 9. 6 8. 8. 7. 9 6 79
.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 ) 0. 8 80
. ( ). ( )( ) ( ) 7.6 Eercise Solve each equation.. ( )( ) 0. ( )( ) 0. ( )( ) 0. 6 0 6. 0. ( 7)( ) 0 7. 0 8. 0 0 9. m m 0.. 9 6. n 8n n 0. 8. p p. w 0w 80w 8
.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
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
.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
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
.. 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
. 8.. 9 6. 7. 87
6. Eercise Find the value of the following rational epressions when and. 7.. 7 8 Find an real number for which each rational epression is undefined.. 7. 7 0 6. 7 9 Simplif each rational epression. 7. 6 8 8. 0 9. 9 9 0.. 8a 8b a b.. 9. 7 8 8. 7 6 6 88