Los Angeles Southwest College. Mathematics Department. Math 115 Common Final Exam. Study Guide (solutions) Fall 2015

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1 Los Angeles Southwest College Mathematics Department Math 5 Common Final Exam Study Guide (solutions) Fall 05 Prepared by: Instr. B.Nash Dr. L.Saakian

2 Chapter. The Real Number System Definitions Place-value chart Millions Thousands Ones Fractions 00,000,000 0,000,000,000,000 00,000 0,000, /0 /00 /,000 /0,000 /00,000 /,000,000 Hundreds Tens Ones Hundreds Tens Ones Hundreds Tens Ones Tenths Hundredths Thousandths Ten- Thousandths Hundred- Thousandths Millionths Numbers. The set or collection,,, 4, 5,, is called the set of natural numbers. The set of natural numbers with additional 0 is called as set of whole numbers. The set of whole numbers with additional negative number is called as set of integer numbers. The set of rational numbers consists of the quotients of two integers. The set of irrational numbers consists of the non-rational numbers represented by a point on the number line (decimals that are not terminating and have no repeating). The set of real numbers consists of the rational and the irrational numbers. Operations Translate words and phrases to a numerical expression Word or Phrase Example Numerical expression Sum of The sum of - and Added to 5 added to More than more than Increased by -6 increased by -6+ Plus plus 4 +4 Difference between The difference between - and -8 --(-8) Subtracted from subtracted from 8 8- From, subtract From, subtract 8-8 Less 6 less Less than 6 less than Decrease by 9 decrease by -4 9-(-4) Minus 8 minus Product of The product of -5 and - (-5)(-) Times times -4 (-4) Twice (meaning times ) Twice 6 (6) Of (used with fractions) ½ of 0 ½(0) Percent of % of -6 0.(-6) As much as / as much as 0 /(0) Quotient of The quotient of -4 and -4/ Divided by -6 divided by 4-6/4 Ratio of The ratio of and /

3 Adding (same sign). Add their absolute values and keep the common sign. Adding (different signs). Subtract their absolute values and keep the sign of the number with the greater absolute value Definition of subtracting: x y x ( y) For adding or subtracting fractions make the same denominator using LCD, add or subtract numerator, and keep the common a c a c denominator b b b Multiplying and dividing real numbers Same signs The product (or quotient) is positive Different signs - The product (or quotient) is negative The multiplying of fractions multiply numerators and keep as a new numerator, multiply the denominators and keep as a new denominator a c b d a c b d Definition of division x x y y Division of whole numbers: 6 R 4 Divident Divisor Quotient Remainder a The quotient b is the number c, if there is one, that when multiplied by b gives a : a b c To get the related multiplication sentence, we use: Dividend Quotient Divisor Property of division: a Any number divided by is that same number: a a a Any nonzero number divided by itself is : a, a 0 Zero divided by any nonzero number is 0: 0 a 0, a 0 Division by zero is not defined. (We agree not to divide by 0): 0 a is undefined or is not defined Exponential notation is , 4 is the exponent, 6 is thebase Division of fractions multiply the first fraction by the reciprocal of the second a c a d b d b c Order of operations: PEMDAS (parentheses, exponents, multiplications or divisions, additions or subtractions) Properties of operations Commutative property: a b b a ab b a Associative property: a ( b c) ( a b) c a ( b c) ( a b) c Distributive property: a ( b c) a b a c Identity: a0 a a a Inverse: a ( a) 0 a a Zero property a0 0 Main property of fraction: a c a b c b Absolute value: x x for x 0 and x x for x 0 Tests for Divisibility: A number is divisible by (is even) if it has an ones digit of 0,, 4, 6, or 8. A number is divisible by 5 if it has an ones digit of 0 or 5. A number is divisible by 0 if its ones digit is 0. A number is divisible by if the sum of its digits is divisible by.

4 A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 6 if its ones digit is 0,, 4, 6, or 8 (is even) and the sum of its digits is divisible by. A natural number that has exactly two different factors, only itself and, is called a prime number. The number is not a prime. A natural number, other than, that is not a prime is composite. Prime numbers from to 57 are:,, 5, 7,,, 7, 9,, 9,, 7, 4, 4, 47, 5, 59, 6, 67, 7, 7, 79, 8, 89, 97, 0, 0, 07, 09,, 7,, 7, 9, 49, 5, 57. Factorization. A number c is a factor of a if a is divisible by c. A factorization of a expresses a as a product of two or more numbers. The prime factorization of a expresses a as a product of two or more prime numbers. Each composite number is uniquely determined by its prime factorization. LCD (Least common denominator) or LCM (least common multiplier):. Factor. Write each number (expression) in prime factored form. List each different denominator. Chose as exponents the greatest exponents from the prime factored forms. 4 Multiply the factors from step to get the LCD or CLM. GCF (the greatest common factor):. Factor. Write each number (expression) in prime factored form.. List common factors.. Chose as exponents the least exponents from the prime factored forms. 4. Multiply the primes from step to get GCF. Fractions and its properties. The main property of the fraction a b is: a / b a c/ b c - we can multiply numerator and denominator of the fraction by any nonzero number. For adding or subtracting fractions make the same denominator using LCD, add or subtract numerator, and keep the common a c a c denominator b b b The multiplying of fractions multiply numerators and keep as a new numerator, multiply the denominators and keep as a new denominator a c b d a c b d Definition of division x x y y, y 0 Division of fractions multiply the first fraction by the reciprocal of the second a c b d a b d c ad bc Mixed numbers have to be transformed to the fractional notation before utilizing any operation. Decimals. To compare two positive numbers in decimal notation, start at the left and compare corresponding digits. When two digits differ, the number with the larger digit is the larger of two numbers. To ease the comparison, extra zeros can be written to the right of the last decimal place. To compare two negative numbers in decimal notation, start at the left and compare corresponding digits. When two digits differ, the number with a smaller digit is the larger of two numbers. Operations with decimals. Addition or subtraction. Lining up the decimal points in order to add or subtract numbers. Multiplication. To multiply using decimal notation: a). Ignore the decimal points, for the moment, and multiply as though both factors are integers. b). Locate the decimal point, so that the number of decimal places in the product is the sum of the number of places in the factors. Count of the number of decimal places by starting at the far right and moving the decimal point to the left. Division. To perform long division by a whole number, place the decimal point directly above the decimal point in the dividend, and divide as though dividing whole numbers. To divide when the divisor is not a whole number, move the decimal point (multiply by 0, 00, and so on) to make the divisor a whole number; move the decimal point the same number of places (multiply the same way) in the dividend; and place the decimal point for the answer directly above the new decimal point in the dividend and divide as if dividing by the whole number. When division with decimals ends, or terminates, the result is called a terminating decimal. If the division does not lead to a remainder of 0, but instead leads to a repeating pattern of nonzero remainders, we have what is called a repeating decimal:

5 Ratio and proportion. A ratio is the quotient of two quantities. The ratio of a to b a is written b or a b. a c When two pairs of numbers have the same ratio, we say that they are proportional. Such an equation is called a proportion. To solve b d a c for a specific variable, equate cross products and then divide on both sides to get that variable b d a c ad alone. Solve for b : bc ad, b. b d c To solve a percent problem, use the percent equation. Amount = percent (as a decimal) * base **************************************************************************************************. Simplify: 8 [ ( )] 8 ( ). Simplify: 8 [9 ( )] 8 7. Simplify: [( 9) ( )] ( ) ( ) ( ) 4 4. Simplify: [ 5 ( 9)] [6 ( )] Find the product and write it in lowest terms: or Find the product and write it in lowest terms: Find the quotient and write it in lowest terms: or or Find the quotient and write it in lowest terms:

6 9. Find the sum and write it in lowest terms: for the numbers5and 5the LCDis Find the sum and write it in lowest terms: for the numbers 4 and 5the LCDis Find the difference and write it in lowest terms: 6 for the numbers6and the LCDis or Find the difference and write it in lowest terms: for the numbers 4and the LCDis Decide whether the statement is true or false: 6 ( ) 6 True 4. Decide whether the statement is true or false: False 5. Decide whether the statement is true or false: 4 ( 5) 4 5 True 6. Decide whether the statement is true or false: False 7. 5( ) (4) Perform indicated operation: [ ( )] 0 0 ( ) ( ) ( ) 6

7 8. Perform indicated operation: ( ) Perform indicated operation: (0.6) (0.8) (.) ( 0.56) 0. For the following word phrase write an expression using x as the variable and simplify. less than the difference between 8 and -5. [8 ( 5)]. For the following word phrase write an expression using x as the variable and simplify. The sum of 4 and 0, increased by. [ 4 ( 0)] 4. For the following word phrase write an expression using x as the variable and simplify. 9 less than the difference between 9 and -. [9 ( )] For the following word phrase write an expression using x as the variable and simplify. The sum of and -7, decreased by 4. [ ( 7)] Otis neglects to keep up his checkbook balance. When he finally balanced his account, he found that the balance was -$.75, so he deposited $ What is his new balance? $.75 $50.00 $ Mike O Hanian owed a friend $8. He repaid $, but than borrowed another $4. What positive or negative amount represents his present financial status? $8 $ $4 $5 $4 $9 6. Peyton Manning of the Indianapolis Colts passed for a gain of 8 yd, was sacked for a loss of yd, and then threw a 4 yd touchdown pass. What positive or negative number represents the total net yardage for the plays? If the temperature drops 7 below its previous level of, what is the new temperature? Evaluate 6x 4 z, if x 5 and z 6 ( 5) 4 ( ) 0 8 7

8 9. Evaluate x 4 y, if x and y 4 ( ) Evaluate 6x 4 z, if x 5 and z 6 ( 5) 4( ) 0 8. Evaluate z (x 8 y), if x 5, y 4, and z ( ) [ ( 5) 8 4] 9( 5 ) 9 ( 47) 4. Combine like terms: 5(5 y 9) (y 6) 5y 45 9y 8 6 y 6. Combine like terms: (r 4) (6 r) r 5 6r 86r r 5 6r r r 86 5 r 4. Combine like terms: p p 8p 6 p 5 p 4 p 8

9 Chapter. Linear Equations and Inequalities in One Variable Properties of equations (inequalities): The Addition property: A B AC B C, A B AC B C, and A B AC B C The Multiplication property: A B AC BC C 0 A B AC BC C 0 and A B AC BC C 0 Solving linear equations (inequalities) in one variable Ax B C ( Ax B C, Ax B C, and so on). Simplify each side separately;. Isolate the variable term on one side;. Isolate the variable (be sure to reverse of the inequality symbol when multiplying or dividing by a negative number); 4. To solve a three-part inequality work with all three expressions at the same time. Three cases for the equation solution: - include variable the equation (inequality) is a solution (the conditional equation); - a false statement there is no solution (the contradiction equation); - a true statement there is infinite number of solutions -any real number is a solution, (the identity equation). Three answers of linear inequalities: - algebraic form x a or x a )7 graph x 6 or ]7 x 6 - interval notation x a (, a) or x a (, a] Ratio and proportion. A ratio is the quotient of two quantities. The ratio of a to b is written a b or a When two pairs of numbers have the same ratio, we say that they are proportional a. b c b d. Such an equation is called a proportion. a c To solve for a specific variable, equate cross products and then divide on both sides to get that variable b d a c ad alone. Solve for b : bc ad, b. b d c Percent notation, n % : n Ratio n % =the ratio of n to 00 n% 00 Fraction notation n% n 00 Decimal notation n% n 0.0 To solve a percent problem using a proportion, we translate as follows: Number N a Amount b Base " Of " translates to " "; " Is" translates to " "; Key words in percent translations: " What " translates to a variable; % translates to " " or " 0.0" 00 9

10 *************************************************************************. Solve the equation and check your solution: 6x 57x x 4 x 8 x 4 xx 4 8 x 4 Check. 6( 4) 5 7( 4) ( 4) True. Solve the equation and check your solution: 4( k 6) (k ) 5 4k 4 k 5 k 6 5 k 5 6 k Check. 4(6) () 5 45 (6 ) Solve the equation and check your solution: (5y 6) ( 4 y) 0 0 True 5y 6 4y 0 y 0 y 0 y 7 Check. (57 6) ( 47) 0 (5 6) ( 8) True 4. Solve the equation and check your solution: ( r) 5( r ) 46r 5r 5 6r 5r5 4 r r Check. ( ) 5( ) ( ) 5 ( 4) True 5. Solve the equation and check your solution: 4xx 7x x Check. 4 9 True 6. Solve the equation and check your solution: 5m 6m m 6 9m 6 m 7 Check True 7. Solve the equation and check your solution:r 5r 6r 68 r 68 r 4 Check True 8. Solve the equation and check your solution: 9 pp 4 4 p 4 p 6 Check. 9 ( 6) ( 6) True 9. Solve the equation and check your solution: 5(m ) 4m 8m 7

11 0m 5 4m 8m 7 6m5 8m 7 6m8m 7 5 m m 6 Check. 5 ( 6) 4( 6) 8( 6) 7 5( ) ( 9) True 0. Solve the equation and check your solution: 6(4x) (x ) 4x 6 4x 6 4x 4x false N / S. Solve the equation and check your solution: (x 4) 6( x ) 6x 6x 6x6x 0 0 true all real numbers. Solve the equation and check your solution: 7r 5r 5r r r 4r r 4r 0 r r Check True. The sum of three times a number and 7 more than the number is the same as the difference between and twice the number. What is the number? x ( x 7) x x x 7 x 4x7 x 4xx 7 6x 8 x The number is equal to 4. During the 09 th Congress ( ), the U.S. Senate had a total of 99 Democrats and Republicans. There were more Republicans than Democrats. How many Democrats and Republicans were there in the Senate? Let x is # of democrats # of republicans is x x ( x) 99 x x 99 x 99 x 88 x 44 # of democrats is 44 and # of republicans is In one day, a store sold 8 5 as many DVDs as CDs. The total number of DVDs and CDs sold that day was 7. How many DVDs were sold? 8 Let x is # of CDs sold # of DVDs is x 5 8 x x x 5 5 x 7 x 05 5 # of CDs is 05 8 # of DVDs is In her job as a mathematics textbook editor, Lauren Morse works 7.5 hr a day. She spent a recent day making telephone calls, writing s, and attending meetings. On that day, she spent twice as much time attending meetings as making telephone calls and spent 0.5 hr longer writing s than making telephone calls. How many hours did she spend on each task? Let x is a time she spent making telephone calls. The time attending meetings is x The time writing e mails is x 0.5 x x x x x 7

12 7 x x.75 4 telephone calls :.75 hr, e mails : hr, meetings :.75.5hr 7. The supplement of an angle measures 0 times the measure of its complement. What is the measure of the angle? Let x is a value of the angle. The value of the supplement angle is 80 x and value of complement angle is 90 x. 80 x0(90 x) 80 x 900 0x 9x 70 x 80 o Measure of the angle is 80 o 8. Find two consecutive odd integers such that when the lesser is added to twice the greater, the result is 4 more than the greater integer. Let x is a value of the lesser odd integer. The value of the greater odd integer is x. ( x ) x ( x ) 4 x 4 x x 4 x4 x 6 x x The numbers are and 9. The perimeter of a certain rectangle is 6 times the width. The length is cm more than the width. Find the width of the rectangle. Let x is a value of the width. The value of the length is x. The perimeter is equals ( width plus length) x ( x ) 6x (x) 6x 4x4 6x x 4 x The width is 0. Two trains are 90 mi apart. They start at the same time and travel toward one another, meeting hr later. If the speed of one train is 0 mph more than the speed of the other train, find the speed of each train. Let x is the speed of the first train. The speed of the second train is x 0. The sum of the distance made by trains in hr is 90 mi x ( x 0) 90 xx x 00 x 50 The speeds of the trains are 50mph and 80mph. The perimeter of a triangle is 96 m. One side is twice as long as another and the third side is 0 m long. What is the length of the longest side? Let x is the length of the second side. The length of the first side is x. The sum of all three sides is 96 m x x0 96 x 66 x The length of the longest side is 44 m. The perimeter of a basketball court is 88 ft. The width of the court is 44 ft less than the length. What are the dimensions of the court? Let x is the length of the court. The width is x 44. [ x ( x 44)] 88 (x 44) 88 4x 88 88

13 4x 76 x 94 The length of the court is 94 ft and the width is 50 ft. Solve the formula d rt for t d r rt r d t r 4. Solve the formula P L W for W PL W W P L 5. Solve the formula M C( r) for r M r r M C C or M C Cr M C Cr M C r C M C r C 5 6. Solve the formula C ( F ) for F 9 9 F C or C F C F C F F C 5 9 F C 5 7. Solve the formula A h( b B) A h ( b B ) for h A h b B x 8 8. Solve the equation 6 4 4x 8 6 4x 08 x 7 9. Solve the equation y 6 y 5 5 ( y ) 5(6 y 5) y 0y 5 y 0y 5 y y 0. If 6 gal of premium unleaded gasoline costs $9.56, how much would it cost to completely fill a 5- gal tank? Let costs of the tank is x The proportion is x 6x x 6 x The costs to fill a tank is $48.90

14 . The distance between Singapore and Tokyo is 00 mi. On a certain wall map, this is represented by in. The actual distance between Mexico City and Cairo is 7700 mi. How far apart are they on the same map? 00 Let distance on the map is x The proportion is 00 x x 00 x x x 5 The distance between Singapore and Tokyo on the map is 5 in.. Solve the inequality 6x x 4x 4 7x 4x 6 7x4x 6 x x (,). Solve the inequality 5( x ) 6x (x ) 4x 5x 5 6x 6x 4x x5 x xx 5 x x 4 [4, ) 4. Solve the inequality 5 x 9 5 x 9 x x 6 [,6] 5. Solve the inequality 5q 6 5q 6 5q 5 q, 5 5 4

15 Linear equation in two variables: Chapter. Linear Equations and Inequalities in Two Variables; Functions Standard form: Ax By C A, B, C are integers, Ais a positive Slope is A C ; x intercept is,0 ; B A y intercept is 0,C B Slope-Intercept form: y mx b m is the slope m rise, (0, b) is the y intercept run Point-Slope form: y y m( x x) m is the slope, ( x, y) is the point on the line set builder notation { x, y Ax B C} or { x, y y mx b} The slope of the line: the slope of the line through two points (, ) (, ) rise y y x y and x y is m run x x Horizontal lines have slope 0: y k. Vertical lines have undefined slope: x k. To find the slope of a line from its equation, solve the equation for y. The slope is the coefficient of x. Parallel and perpendicular lines: Parallel slopes are equal; Perpendicular slopes are negative reciprocals (flip over and negative). Graphing Linear equations in two variables:. Find at least two ordered pairs that satisfy the equation (make T-bar);. Plot the corresponding points.. Draw a straight line through the points. Graphing linear inequality in two variables:. Graph the line that is the boundary of the region.. Make the line solid if the inequality is or and dashed if the inequality is or.. Use any point not on the line as a test point. Substitute for the x and y in the inequality. If the result is true, shade the region of the line containing the test point. If the result is false, shade the other region. Introduction to Functions. In an ordered pair ( x, y, ) x and y are called the component of the ordered pair. A function is a set of ordered pairs ( x, y ) in which each first component corresponds to exactly one second component. If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function. The notation f ( x ) means the value of x for the function f. It represents the y value that corresponds to x. The domain of a function is the set of numbers that can replace x in the expression for the function. The range is the set of y values that result as x is replaced by each number in the domain. *************************************************************************. Find the x -intercept and y -intercept: xy 4 x 0 y 4 y 8 y 0 x 4 x x intercept is (,0) and y intercept is (0, 8). Find the x -intercept and y -intercept: 5xy 0 x 0 y 0 y 0 5

16 y 0 5x 0 x 4 x intercept is (4,0) and y intercept is (0, 0). Find the x -intercept and y -intercept: y.5 0 y.5 none;(0,.5) x intercept is none and y intercept is (0,.5) 4. Find the x -intercept and y -intercept: x 4 0 x 4 (4,0);none x intercept is (4,0) and y intercept is none 5. Find the slope of the line through a pair of points: (4, ) and (, 8) y y m x x 8 ( ) m 8 7 m m 7 slope is equal to 4 ( 4) Find the slope of the line through a pair of points: ( 8,0) and (0, 5) y y m x x 5 0 m 5 m 0 ( 8) 8 5 m 8 5 slope is equal to 8 7. Find the slope of the line through a pair of points: ( 8,6) and ( 8, ) y y 6 ( 6) m m m 7 m scope is underfind x x 8 ( 8) 8 ( 8) 0 8. Find the slope of the line through a pair of points: (6, 5) and (, 5) y y 5 ( 5) m m 0 m m 0 slope is equal to 0 x x For the pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither parallel nor perpendicular. x 5 y 4 4x0y 4 5y x 4 y x m m m two lines are parallel 0y 4x y x m For the pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither parallel nor perpendicular. x y 6 xy y x 6 y x m m m two lines are perpendicular y x y x m. For the pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither parallel nor perpendicular. 8 x 9 y 6 8x 6y 5 6

17 8 8 9y 8x 6 y x m y 8x 5 y x m 6 m m and m m two lines are neither parallel nor perpendicular. For the pair of equations, give the slopes of the lines and then determine whether the two lines are parallel, perpendicular, or neither parallel nor perpendicular. 5 xy x 5y 0 y 5x y 5x m 5 m 5y x 0 y x m m two lines are perpendicular 5 5. Write an equation for the line passing through the given point and having the given slope. Give the final answer in the slope - intercept form: (4,), m y y m( x x ) y ( x 4) y x 8 yx 7 4. Write an equation for the line passing through the given point and having the given slope. Give the final answer in the slope - intercept form: (,5), m y y m( x x ) 4 y 5 [ x ( )] y 5 x 9 y x 5. Write an equation for the line passing through the given point and having the given slope. Give the final answer in the slope - intercept form: (,), m 4 y y m( x x ) y 4[ x ( )] y 4x 4 y 4x 6. Write an equation for the line passing through the given point and having the given slope. Give the final answer in the slope - intercept form: (,7), m y y m( x x ) y 7 ( x ) y 7 x 6 yx 7. Write an equation for the line passing through the given pair of points. Give the final answer in the slope - intercept form. (8,5) and (9, 6) y y 6 5 m m m m x x 9 8 y y m( x x ) y 5 ( x 8) y 5 x 8 y x 7

18 8. Write an equation for the line passing through the given pair of points. Give the final answer in the slope - intercept form. (4,0) and (6,) y y 0 m m m m x x 6 4 y y m( x x ) y 0 ( x 4) y 0 x 4 yx 6 9. Write an equation for the line passing through the given pair of points. Give the final answer in the slope - intercept form. (, ) and (, 4) y y 4 ( ) m m m m x x ( ) 5 5 y y m( x x ) 6 y ( ) [ x ( )] y x y x Write an equation for the line passing through the given pair of points. Give the final answer in the slope - intercept form. ( 4,0) and (0,) y y 0 m m m m x x 0 ( 4) 4 y y m( x x) y 0 [ x ( 4)]. Graph the linear inequality: x5y 9 To graph the boundary, which is the line x5y 9, find its intercepts. x 5y 9 x 5y 9 let y 0 let x 0 x y 9 x9 5y 9 x 9 y 5 The x intercept is (,0) and y intercept is 9 0,.draw a dashed line through these points. In 5 order to determine which side of the line should be shaded, use (0,0) as a test point. Substituting 0 for x and y will result in the inequality0 9, which is False. Shade the region not containing the origin. The dashed line shows that the boundary is not part of the graph. y x 8

19 . Graph the linear inequality: x y 6 To graph the boundary, which is the line x y 6, find its intercepts: x y 6 x y 6 let y 0 let x 0 x y 6 x 6 y 6 x y The x intercept is (,0) and y intercept is (0, ).draw a dashed line through these points. In order to determine which side of the line should be shaded, use (0,0) as a test point. Substituting 0 for x and y will result in the inequality 0 6, which is true. Shade the region containing the origin. The dashed line shows that the boundary is not part of the graph.. Graph the linear inequality: xy 0 The equation of the boundary is xy 0. This line goes through the origin, so both intercepts are (0,0). Second point on this line is (,). Draw a solid line through (0,0) and (,).Because (0,0) lies on the boundary, we must choose another point as the test point. Using (0, ) results in the inequality 6 0, which is false. Shade the region not containing the test point. The solid line shows that the boundary is part of the graph. 4. Give the Domain and Range and decide whether each relation is or is not a function {(-4,),(-,),(0,5),(-,-8)} Domain: (-4,-,0), Range: (,,5,-8), Not a function 5. Give the Domain and Range and decide whether each relation is or is not a function {(,7),(,4),(0,-),(-,-),(-,5)} Domain: (,,0,-,-), Range: (7,4,-,-,5), Function 9

20 The Addition property: A B then A C B C Chapter 4. System of Linear Equations and Inequalities Properties of equations: The Addition property: A B and C D then AC B D The Multiplication property: A B then AC BC C 0 The substitution property: A B and A C then C B The system of two equations in two variables: Solving the system in two variables by using elimination (by substitution or addition properties of equations): ax by c ax by c a( dy f ) by c or ( b d) y c f x dy f and solve the linear equations in one variable after that. ax dy f Three cases for solutions of the system:. The solution gives the (single) ordered-pair solution of the system => the system is consistent and the equations are independent.. The solution gives a parallel lines or false statement => the system is inconsistent and the equations are independent.. The solution gives the same line or true statement => the system is consistent and the equations are dependent. There is an infinite number of { x, y Ax B C} or { x, y y mx b} where the equation is one of the original solutions, that could be written in set-builder notation as equations. Solving systems of linear inequalities:. Graph each inequality on the same axes.. Choose the intersection. The solution set of the system is formed by the overlap of the regions of the two graphs. Solving an Applied Problem with Two Variables: Step. Read the problem carefully. What information is given? What are you asked to find? Step. Assign variables to represent the unknown values. Use a sketch, diagram, or table, as needed. Write down what each variable represents. Step. Write two equations using two variables. Step 4. Solve the system of two equations. Step 5. State the answer. Label it appropriately. Does it seem reasonable? Step 6. Check the answer in the words of the original problem. *************************************************************************. Decide whether the given ordered pair is a solution of the given system. (,4) 4xy 4 5x y 9 4() (4) true 5() true The ordered pair (,4) is a solution of the given system. Decide whether the given ordered pair is a solution of the given system. ( 5,) x 4y xy4 ( 5) 4() 5 8 true 0

21 ( 5) () false The ordered pair ( 5,) is not a solution of the given system. Solve the system by graphing: x y4 x y 5 To graph the equations, find the intercepts. x y 4; let y 0; then x 4 let x 0; then y 4 Plot the intercepts, (4, 0) and (0, 4), and draw the line through them x y 5; let y 0; then x.5 let x 0; then y 5 Plot the intercepts, (.5, 0) and (0, 5), and draw the line through them. It appears that the lines intersect at the point (,). Check this by substituting for x and for y in both equations. Since (,) satisfies both equations, the solution set of this system is {(,)} 4. Solve the system by graphing: xy 4 x y To graph the equations, find the intercepts. x y 4; let y 0; then x 4 let x 0; then y Plot the intercepts, (4, 0) and (0, ), and draw the line through them x y ; let y 0; then x let x 0; then y Plot the intercepts, (, 0) and (0, ), and draw the line through them It appears that the lines intersect at the point (0, ). Check this by substituting 0 for x and for y in both equations. Since (0, ) satisfies both equations, the solution set of this system is {(0, )} 5. Solve the system by the substitution method. x y 9 x y 8 x y 8 x y8

22 x y 9 ( y 8) y 9 5y 4 9 5y 5 y x y 8 x 8 x 7 {(7,)}; the ordered pair (7,) 6. Solve the system by the substitution method. x y 0 x y 0 y x 4xy is a solution of the given system 4x ( x) 4x 4x 0 false n/ s ; 7. Solve the system by the substitution method. x y x y x x 4x x y x y x y y 9 {(,9)}; the ordered pair (,9) 8. Solve the system by the substitution method. y 4 6 x y 7 x x y7 x y 7 x (7 x) 7 x 7 x true 9. Solve the system by the elimination method. 5 x y 5 x y x y x y x x is a solution of the given system There are infinity solutions {( x, y) x y 7} x y y y y {(, )}; the ordered pair (, ) is a solution of the given system 0. Solve the system by the elimination method. xy x y 4xy4 x y x y x y 7x x x y y 6 y y 6 y 6. Solve the system by the elimination method. x y x y 4 4 x y x y 9x6y 9 8x 6y x y y 0 y 5 {(, 6)}; the ordered pair (, 6) is a solution of the given system x y 4x y {(,5)}; the ordered pair (,5) is a solution of the given system. Solve the system by the elimination method. 5xy 5x y 0x 4y 6 0x4y 5 0x4y 5 0 x4y 5 0 false n/ s ;

23 . Bill Kunz went to the post office to stock up on stamps. He spent $9.44 on 56 stamps, made up of a combination of 9-cent and 4-cent stamps. How many stamps of each denomination did he buy? let x is amount of 9 cent stamps and y is amount of 4 cent stamps 4x 4y 44 x y56 4 x y 56 9 x4y944 9x4y 944 9x4y944 5x 600 x40 x y y 56 y 6 # of 9 cent stamps is 40; and # of 4 cent stamps is 6 4. A 40% dye solution is to be mixed with a 70% dye solution to get 0 L of a 50% solution. How many liters of the 40% and 70% solutions will be needed? let x is volume of the 40% dye solution and y is volume of the 70% dye solution x y x 0.7 y x 4y 480 x y 4x7y 600 4x7y600 y 0 y 40 x y 0 x 40 0 x 80 The volume of the 40% dye solution is 80 L; and the volume of the 70% dye solution is 40L 5. Two trains start from towns 495 mi apart and travel toward each other on parallel tracks. They pass each other 4.5 hr later. If one train travels 0 mph faster than the other, find the speed of each train. let x is the speed of the first train and y is the speed of the second train 4.5x 4.5y 45 x y0 4.5 x y0 4.5x4.5 y x4.5y x4.5y495 9y 450 y 50 x y 0 x 50 0 x 60 y 6 y 6 The speed of the first train is 60 mph; the speed of the second train is 50 mph 6. If a plane can travel 440 mph into the wind and 500 mph with the wind, find the speed of the wind and the speed of the plane in still air. let x is the speed of the plane and y is the speed of the wind xy440 x y 440 x y500 x y y 440 y 0 y 0 x y 500 x 940 x470 The speed of the plain in still air is 470 mph; the speed of the wind is 0 mph 7. Nancy Johnson invested $8,000. Part of it was invested at % annual simple interest, and the rest was invested at 4%. Her interest income for the first year was $650. How much did she invest at each rate? let x is the first investment, the second is 8000 x x 4(8000 x) x% (8000 x) 4% x7000 4x x 7000 x 7000 She invested $7,000 at % and $,000 at 4%

24 8. Graph the solution set of the system of linear inequalities. x y x y4 Graph x y as a solid line through its intercepts, (, 0) and (0, ). Using (0,0) as a test point will result in the false statement 0,so shade the region not containing the origin. Graph x y 4 as a solid line through its intercepts, (4, 0) and (0, 4). Using (0,0) as a test point will result in the true statement 0 4,so shade the region containing the origin. The solution set of this system is the intersection of the two shaded regions, and includes the portions of the two lines that bound this region. 9. Graph the solution set of the system of linear inequalities. y xy6 Graph y x as a solid line through its intercepts, (0, 0) and (, ). This line goes through the origin, so a different test point must be used. Choosing ( 4,0) as a test point will result in the true statement 0 8, so shade the region containing ( 4, 0). Graph xy 6 as a solid line through its intercepts, (, 0) and (0, ). Using (0,0) as a test point will result in the true statement 0 6, so shade the region containing the origin. The solution set of this system is the intersection of the two shaded regions, and includes the portions of the two lines that bound this region. 4

25 Chapter 5. Exponents and Polynomials Definitions and rules: A term is called a monomial if there is no division by a variable expression A polynomial is an algebraic expression made up with a term or a finite sum of terms with real coefficients and whole number exponents The degree of a term is the sum of the exponents of the variables. If a 0, then for inters m and n, the following are true Product rule m m n m a a a Power rule m n mn m m m m m ( a ) a, ( ab) a b, ( a ) a b m b Zero exponents 0 a ( a 0) n Negative exponent a ( a 0) n a m Quotient rule a m n a ( a 0) n a Negative. to positive. Rules a b, a b m n m m n m b a b a Adding polynomials add likes terms (place like terms in columns so they can be added) 4x 6x8 x 4x x x5 Subtracting polynomials change the signs of the terms in the second polynomial and add after that. Multiplying polynomials multiply each term of the first polynomial by each term of the second polynomial. Then add like terms (place like terms x x 4x in columns so they can be added) x 5 5x 0x 0x 5 4 x 6x x x 4 x x x 7x 5 a b a b Dividing a polynomial by a Monomial c c c Dividing polynomials - x x x 4x 4x 5x 8 (4x x ) ( x x) 4x 8 (4x ) 6 x 5x 8 Divident 4 x 4 x 5 x 8 6 x x Remainder Divisor x x Divisor Formulas : ( a b)( a b) a b, ( a b) a ab b *************************************************************************. Simplify: ( x y z) ( yz ) 5

26 6 5 0 x y z y z 7x y z. Simplify: 5 ab 8 c 6 5. Simplify: 6 x y 0 z 4 6 5a b 6 c 5 6x y 5 z 9 4 5a b 8 c x y 0 z 6 4. Simplify: ( r s) ( r s ) 4 5 ( ) r s ( ) r s ( ) r s 8 7 r s 5. Simplify. Use only positive exponents: (6) x z x (6) z 9 6. Simplify. Use only positive exponents: xy x y xy 7. Simplify. Use only positive exponents: ( ) r s ( ) r s ( ) r s (6 x z ) 5 ( xy ) x y ( r s) ( r s ) x 6z x y r s mn p mn p 8. Simplify. Use only positive exponents: 4 4 m np m np m n p ( m n p ) 9. Perform the operation: 4 9a a 4 4a 4a 4 a 6a 4 a a 0. Perform the operation: 8m 7m m 7m 6 5m 4m (9a a ) (4a 4a ) ( a 6a ) (8m 7 m) (m 7m 6) a a 4 5m 4m 6 6

27 . Perform the operation: (6b c) ( b 8 c) 6b c b8c 4b 5c. Perform the operation: (4x xy ) ( x xy 4) 4x xy x xy 4 6x xy 7. Find the product: ( m7)( m 5) m m m Find the product: (m n)( m 5 n) m 0mn mn 5n 4b 5c 6x xy 7 m m 5 m 7mn 5n 5. Find the product: p (p 5 p)( p p ) 5 4 (6 p 5 p )( p p ) 6 p p 6 p 5p 0 p 5 p p 5 p p 6 p 5 p Find the product: (a ) (a)(a )(a ) 8a 4a 8a 4a a (4a 4a )(a ) 8a a 6a 7. Find the product. Use special product formulas to simplify: ( a8)( a 8) 8. Find the product. Use special product formulas to simplify: m m ( m ) a m 64 4m 4 9. Find the product. Use special product formulas to simplify: (5y x)(5y x) (5 y) ( x) 5y 9x 0. Find the product. Use special product formulas to simplify: z z5 5 ( z 5) z 0z 5. Find the product. Use special product formulas to simplify: ( r) r 5 t (5 t) (r 5 t) 4r 0rt 5t. Find the product. Use special product formulas to simplify: (6m5)(6m 5) (6 m) (5) 6m 5 7

28 . Find the product. Use special product formulas to simplify: p(p 7)(p 7) p[( p) (7) ] p9p 49 9 p 49 p 4. Find the product. Use special product formulas to simplify: 4 ( x) x x x 5. Find the product. Use special product formulas to simplify: [(4 r) 4r () ] (6r 6r 4) (4r ) 6r 6r 4 6. Perform the division: 8t 4t 4t t t t 5 8t 4t 4t t t t t 5 4 0m 0m 5m 7. Perform the division: 5m 5 4 0m 0m 5m 4m m 5m 5m 5m 8. Perform the division: 0m n 5m n 6m n m n m n m n 9. Perform the division: r r r r 5r 6r 5 (r 6 r ) r 6r5 ( r r) r 5 ( r 9) 6 ( ) m n m n m n m n r 5r 6r 5 r 4m m 6 m n mn n 5 r r r 6 0. Perform the division: 4x 5 4x5 6x 5 (6 x 0 x) 0x 5 ( 0x 5) 0 6x 5 4x 5 8 4x 5

29 . Perform the division: r 4 r r r 5 r 4 ( r r ) 5 ( r ) 4 5r r r 4 r r 4 9

30 Chapter 6. Factoring and Applications Distributive property: a ( b c) a b a c Grouping property: a b a c a ( b c) GCF (the greatest common factor):. Factor. Write each number (expression) in prime factored form.. List common factors.. Chose as exponents the least exponents from the prime factored forms. 4. Multiply the primes from step to get GCF. Factoring Expressions: Factoring fournomials(double grouping): Step. Group terms. Collect the terms into two groups so that each group has a common factor. Step. Factor within groups. Step. Factor the entire polynomial. Factor out a common binomial factor from the results of step. Step 4. If necessary, rearrange terms. If step does not result on a common factor, try a different grouping. Factoring trinomials: mnac ax bx c ax mx nx c duble grouping ( ax mx) ( nx c) mnb Factoring binomials (use formulas) : a b ( a b)( a b), a b ( a b)( a ab b ), a ab b ( a b) Zero-factor property of multiplication: if ab 0 then a 0 or b 0 Quadratic equation in one variable ax bx c 0 Solving the quadratic equations by using the zero-property of multiplication by factoring the quadratic equation and solving the linear equations after that: x 4x 0 ( x )( x ) 0 x 0 or x 0 x or x, *************************************************************************. Factor completely: GCF 8m n 8m n 4m n 8m n n 8m n 8 m n ( n ). Factor completely: m mp mr pr 5m 5mp mr 6 pr 5 m( m p) r( m p) 5m r ( m p)(5m r). Factor completely: 6m 4m p 4mp p 6m 4m p 4mp p 4m p 4 m (4 m p ) p(4 m p ) (4 m p )(4 m p) 4. Factor completely: GCF 9 p q 6 p q 45p q 8p q p q 4 p 9 p q 5p q 9p q 9q 0 9 p q(4 p 5 p q 9 q)

31 5. Factor completely: 6. Factor completely: GCF 5 7. Factor completely: GCF mn 8. Factor completely: GCF ( a b) r ra a ( r a)( r a) 5y 5y 0 5( y y 6) 5( y)( y ) m n m n mn mn( m mn n ) mn( m n)( m n) ( a b) x ( a b) x ( a b) ( a b)( x x ) ( a b)( x 4)( x ) 6x 7x 6x 8x 9x 6x 8x 9x x 9. Factor completely: x(x 4) (x 4) (x4)(x ) 0. Factor completely: t t 8 prime. Factor completely: s st 5t s 5st 4st 5t s 5st 4st 5t s s(4s 5 t) t(4s 5 t) (4s 5 t)( s t). Factor completely: 8 65x 7x 8 6x x 7x 8 6x x 7x 9 x 9( 7 x) x( 7 x) ( 7 x)(9 x) t. Factor completely: 4 x ( x )( x ) ( x )( x )( x ) 4. Factor completely: GCF 8 a 8 8(4a ) 8[( a) ] 8(a)( a ) 5. Factor completely: x 0x 5 x x5 5 ( x 5) 6. Factor completely: x 4x 7 GCF ( x x 6) ( x x 6 6 ) ( x 6) 7. Factor completely: ( t) (4 s ) 7t 64s 6 (t 4 s )[( t) ( t)(4 s ) (4 s ) ] 4 (t 4 s )(9t ts 6 s )

32 8. Factor completely: 5t 8s (5 t) ( s ) 6 (5t s )[(5 t) (5 t)( s ) ( s ) ] 4 (5t s )(5t 0ts 4 s ) 9. Factor completely: 6r 5a (4 r) (5 a) (4r 5 a)(4r 5 a) 0. Factor completely: 8w 6 prime. Solve the equation: (x 7)( x x ) 0 (x 7)( x )( x ) 0 x 7 0 x 7 x x 7 0 x 7 x x 0 x,,. Solve the equation: x x x x x 4 4 x 0 x x ( x ) ( x ) x x 0 ( x )( x) 0 x 0 x {, }. Solve the equation: x x x x x x 0 GCF x x( x x ) 0 x( x )( x ) 0 x 0 x 0 x x 0 x {,0,} 4. Solve the equation: 6r 9r 0 GCF r r(6r 9) 0 r[(4 r) ] 0 r(4r )(4r ) 0 r 0 4r 0 r 4 4r 0 r 4 0,, A certain triangle has its base equal in measure to its height. The area of the triangle is7 m. Find the base and height measure. baseheight let x is the base. The area of triangle is a

33 x x x a 7 x 44 x 44 0 ( x )( x ) 0 x 0 x x 0 x The base is m : hegative solution does not make sense, since x represents the lenght, which cannot be negative. 6. The product of the second and third of three consecutive integers is more than0 times the first integer. Find the integers. Let the first conecutive integer is x, the second x, and the third x ( x )( x ) 0x xx7 0 x 0 x 7 0 x 7 x x x x 0 x 7x 0 Three consecutive integer numbers are : 0,, or 7,8,9 7. A ladder is leaning against a building. The distance from the bottom of the ladder to the building is 4 ft less than the length of the ladder. How high up the side of the building is the top of the ladder if that distance is ft less than the length of the ladder. Let the length of the ladder is x ( x 4) ( x ) x ( x0)( x ) 0 x0 0 x 0 x 0 x x 8x 6 x 4x 4 x x x 0 0 The length of the ladder is 0 ft, and the high of the building is 8 ft. The solution ft give a negative value of distance from the bottom of the ladder to the building, that does not make sence. 8. An object projected from a height of 48 ft with an initial velocity of ft per sec after t seconds has height h 6t t 48 ( a) After how many seconds is the height 64 ft? ( b) After how many seconds does the object hit the ground? a t t ) t 0 t 6( t t ) 0 Solution sec b t t ) t t 48 0 t 0 t t 0 t t t 0 6t t ( t t ) 0 ( t ) 0 t 6t t 6 0 t 0 ( t )( t) 0 The solution is sec. The negative solution,, does not make sence, since t represents time, which cannot be negative.

34 Chapter 7. Rational Expressions and Applications A rational expression is the quotient of two polynomials with denominator not 0. The fundamental property of rational expression: to find the value(s) for which a rational expression is undefined, set the denominator equal to 0 and solve the equation. Writing a rational expression in lowest term: - Factor numerator and denominator; - Use the fundamental property to divide out common factors. Multiplying and dividing rational expressions: - Factor numerators and denominators completely; - Note the operation. If the operation is division, use the definition of division to rewrite as multiplication by the reciprocal; - Write in the lowest terms, using the fundamental property. Adding and subtracting rational expressions: - Find the LCD; - Rewrite each rational expression with the LCD as denominator; - Add the numerators to get the numerator of the sum. The LCD is the denominator of the sum. - Write in the lowest term. Complex fraction: - Method - simplify the numerator and denominator separately; then divide the simplified numerator by simplifying denominator; - Method Multiply the numerator and denominator of the complex fraction be the LCD of all the denominators in the complex fraction. Write in lowest term. Solving equations with rational expressions: - Multiply each side of the equation by the LCD to clear the equation of fractions. Be sure to distribute to every term on both sides. - Solve the resulting equation. - Check each proposed solution. *************************************************************************. Write the rational expression in lowest terms: 7t 4 7t 5t 4t 0 7t 4 7t t0 7t 4 7 t( t 5) 4( t 5) ( t5)(7t4) 7t 4 7t 4 t 5. Write the rational expression in lowest terms: ( x5)( x) ( x5)( x). Write the rational expression in lowest terms: 5k 5k 5k k 6 5k 5 k( k ) ( k ) 5k 4. Write the rational expression in lowest terms: x x x5 6x5 5k k6 5k ( k)(5k) 5k x x5 x 7x5 x x k 4

35 x x 5x x 5 x 5x x 5 x x 5x x 5. Multiply: x x x x 6x x x x x ( x )(x )( x ) ( x)( x) x(x 5) ( x 5) (x5)( x) x(x 5) ( x 5) (x5)( x) x( x ) ( x ) x x x x x (x)( x) x 6. Multiply: 7. Divide: k k 4k 5k 6k 7k k k k 4k k 4k k 4k 4k k 6k 4k k ( k )( k ) k ( k )(k )( k )(4k ) (k )(k )( k )( k ) m mp p m 4mp p m mp p m mp 8p ( m p)( m p) ( m p)( m p) ( m p)( m p) ( m p)( m 4 p) k( k ) ( k ) 4 k( k ) ( k ) k(k ) (k ) ( k )( k ) ( m p)( m p) ( m p)( m 4 p) ( m p)( m p) ( m p)( m p) 4k k m 4 p m p 8. Divide: 4 ( q ) ( q ) q 6q 9 q q q q 4 4 ( q ) ( q ) ( q ) ( q )( q ) ( q ) 4 ( q ) ( q ) ( q ) ( q )( q ) ( q ) 4 ( q) ( q) ( q ) 4m m 9. Add: m m m 6m 5 4m m ( m )( m ) ( m 5)( m ) LCD ( m )( m )( m 5) 4 m( m 5) (m )( m ) ( m )( m )( m 5) ( m )( m )( m 5) 4m 0m m 4m m ( m )( m )( m 5) 4m 0m m 4m m ( m )( m )( m 5) ( m )( m )( m 5) 6m m ( m )( m )( m 5) a 4a 0. Add: a a 4 a 7a 5

36 a 4a ( a 4)( a ) ( a 4)( a ) LCD ( a 4)( a )( a ) a( a ) 4 a( a ) a a 4a 4a ( a 4)( a )( a ) ( a 4)( a )( a ) ( a 4)( a )( a ) ( a 4)( a )( a ) a a 4a 4a ( a 4)( a )( a ) 5a a a(5a) ( a 4)( a )( a ) ( a 4)( a )( a ) x z x z. Perform indicated operation: x xz 0z x 4z x z x z x z x z x 5xz 4xz 0 z x ( z) x(x 5 z) z(x 5 z) ( x z)( x z) x z x z x z (x 5 z)( x z) ( x z)( x z) LCD (x 5 z)( x z)( x z) ( x z)( x z) ( x z)(x 5 z) (x 5 z)( x z)( x z) (x 5 z)( x z)( x z) x 4xz xz z x 5xz xz 5z (x 5 z)( x z)( x z) (x 5 z)( x z)( x z) x xz z x 7xz 5z (x 5 z)( x z)( x z) (x 5 z)( x z)( x z) x xz z x 7xz 5z (x 5 z)( x z)( x z) z 4xz 7z (x 5 z)( x z)( x z) z(4x 7 z) (x 5 z)( x z)( x z). Perform indicated operation: 6 k k k k k k 6 k k k k ( k )( k ) k k LCD k( k )( k ) 6( k ) ( k ) k k( k )( k ) k( k )( k ) k( k )( k ) 6k 6 k k k( k )( k ) k( k )( k ) k( k )( k ) 6 k( k ) k( k ) ( k )( k ) 6k 6 k k 7k 9 k( k )( k ) k( k )( k ) x. Simplify: x x 8 6

37 8x x 8x LCD 8x x x 8x 8 8 8x xx ( ) 8( x ) xx ( ) 8 x 4. Simplify: m m LCD m ( m) ( m) m ( m) ( m) m m m m m 5. Simplify: m m m m LCD ( m )( m )( m ) ( m )( m )( m ) ( m )( m )( m ) ( m) ( m) ( m )( m )( m ) ( m )( m )( m ) ( m) ( m) ( m )( m ) ( m )( m ) ( m )( m ) ( m )( m ) m m 6 m 8m 6 m 8m 6 m m m m 9 m 9m8 mm ( ) ( m)( m8) 6. Simplify: 5 p 7. Solve the equation and check your solutions: p p p p p ( p ) ( p ) ( p )( p ) p p ( p )( p ) ( p )( p ) ( p )( p ) p p p p p p ( p )( p ) ( p )( p ) ( p )( p ) ( p )( p ) ( p )( p ) p p p p ( p )( p ) ( p )( p ) ( p )( p ) ( p )( p ) ( p )( p ) ( p )( p ) ( ) 6 p p p Check : ( ) ( ) ( ) 9 4 7

38 true The solution set is { } k 4 8. Solve the equation and check your solutions: 5 k4 k4 k 5( k 4) 4 k 5k 0 4 k5k0 4 k 4 k 4 k 4 k 4 k 4 k 4 k4 k4 4k 0 4 ( k 4) ( k 4) 4k 0 4 4k 6 k 4 k4 k4 4 4 The proposed solution, 4, makes an original denominator equal 0, Check : so is not a solution. The solution set is 9. 0 Solve the equation and check your solutions: z 5 z 5 z 5 0 ( z 5) ( z 5) 0 z 5 z 5 z 5 ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) z0 z5 0 z0 z5 0 ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) z 5 0 ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) ( z 5)( z 5) z 5 0 z 5 0 Check : ( 5) 5 The proposed solution, 5, makes an original denominator equal 0, so is not a solution. The solution set is x4 5 x4 0. Solve the equation and check your solutions: x x x 4x x 5x 6 x4 5 x4 ( x )( x ) ( x )( x ) ( x )( x ) ( x 4)( x ) 5( x ) ( x 4)( x ) ( x )( x )( x ) ( x )( x )( x ) ( x )( x )( x ) ( x 4)( x ) 5( x ) ( x 4)( x ) ( x )( x )( x ) ( x )( x )( x ) 8 x x x x x x x ( x )( x )( x ) ( x )( x )( x ) x 4x x 5x 4 ( x )( x )( x ) ( x )( x )( x ) ( x )( x )( x ) ( x )( x )( x ) x 4x x 5x 4 x 6 : Check true The proposed solution, 6, does not make an original denominator equal 0, so it ' s a solution. The solution set is {6}

39 . One-third of a number is more than one-sixth of the same number. What is the number? Let the number is x x x 6 LCD 6 6 x 6 x 6 xx x The number is. A boat can go 0 mi against a current in the same time that it can go 60 mi with the current. The current is 4 mph. Find the speed of the boat in still water. Let the speed of the boat in the still water is x 0 60 x4 x4 LCD ( x 4)( x 4) 0 60 ( x 4)( x 4) ( x 4)( x 4) 0( x 4) 60( x 4) x4 x4 0x80 60x 40 40x 0 x 8 The speed of the boat in the still water is 8 mph. Working alone, Jorge can paint a room in 8 hr. Catarina can paint the same room working alone in 6 hr. How long will it take them if they work together? Let x the number of hours it takes Jorge and Catarina to paint a room, working together x x LCD 4 4 x x 4 x4x 4 x 4 x x Working together, Jorge and Catarina can paint a room in hr One pipe can fill a swimming pool in 6 hr, and another pipe can do it in 9 hr. How long will it take the two pipes working together to fill the pool 4 full? Let x the number of hours it takes two pipes to fill a pool full, working together 4 x x LCD 6 6 x x 6 6x4x 7 0x 7 x Working together, two pipes can fill a pool full in hr

40 Chapter 8-9. Roots and Radicals. Quadratic Equations Radicals. Product rule: n a n b n ab ; Quotient rule: n n a b n a b Rationalizing the denominator: the denominator of a radical can be rationalized by multiplying both the numerator and denominator by a number that will eliminate the radical from the denominator. If the radical expression contains two terms in the denominator and at least one of those terms is a square root radical, multiply both numerator and denominator by the conjugate of the denominator. m n n m n Using rational numbers as exponents: Solving equations with radicals: - Isolate the radical; - Square each side; - Combine like terms; - If there is still a term with a radical, repeat steps -. - Solve the equation for proposed solutions. - Check all proposed solutions in the original equation. m n n n a a; a a a ; a m a m n The Squaring property of equation: A B then A B and all solutions of the original equation are among the solutions of the squated equation. The Square root property of equation: if k is a positive and if x k, then x k or x k,, The solution set is k k which can be written k Quadratic equation in one variable ax bx c 0 Solving the quadratic equations by using the quadratic formula a ax bx c 0 x, b b 4ac *************************************************************************. Find the square root: Find the square root: Find the distance between the pair of points: (5,7) and (,4) ( x x ) ( y y ) ( 5) (4 7) 40 ( 4) ( ) Find the distance between the pair of points: (, 6) and ( 4, 0)

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