Course 15 Numbers and Their Properties

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1 Course Numbers and Their Properties KEY Module: Objective: Rules for Eponents and Radicals To practice appling rules for eponents when the eponents are rational numbers Name: Date: Fill in the blanks. When multipling epressions with eponents, ou should add the eponents if the bases are the same. The term is the same as (). false (true, false) An quantit raised to the zero power is equal to. A non-zero number raised to a negative eponent is equal to over the number raised to a(n) positive eponent. An quantit raised to the one-half power is equal to the square root of the quantit. Problem Set: Simplif each epression.. () 6. /. (r ) r. (n) n. (9a ) (a ) 8 6a 6. (t) 0 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafah_k. of

2 Course Numbers and Their Properties b 7. 6 b 8 b 8 8. or z z ( ) a b c 6 a b c ( ) b. /8 / 9/8. ( bc ) ( a b c) a 9 8ac 6 z. z z / ( 8 ) 6. 0 ( 6 ) / Reflection: Eplain the difference between and (). is equal to, while () is equal to which is the same as or PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafah_k. of

3 Course Numbers and Their Properties Module: Rationalizing the Denominator in Rational Epressions KEY Objective: To practice rationalizing the denominator in rational epressions Name: Date: Fill in the blanks. is the same as the square root of. An nonzero number raised to the zero power is equal to one. m n b is the same as taking the th n root of m b. Rationalizing the denominator involves taking a rational epression that has a radical in the denominator and converting it to an equivalent epression that does not have a radical in the denominator. To rationalize the denominator, multipl the numerator and denominator b a factor that changes the denominator from a(n) irrational number to a(n) rational number. Problem Set: Simplif the following a b 8 9a b 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafbh_k. of

4 Course Numbers and Their Properties a bc ( ) ( ) ( ) 8 ( a ) ( b ) ( c ) 6 9 ( ) ( ) 6 a b c b a c Rationalize the denominator a b. 6a b. 6ab 8ab 6 ab a * ab ab 6 a b ba a 8 ab ab 8 a b ab ab ab a b ab a a ba ba 8ab 8 ab b Reflection: Eplain in our own words what ou are doing when ou rationalize the denominator. Rationalizing the denominator is taking a rational epression that has a radical in its denominator and converting it to an equivalent epression that is without a radical in its denominator. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafbh_k. of

5 Course Numbers and Their Properties Module: Appling Rules for Eponents and Radicals KEY Objective: To practice simplifing rational epressions with eponents and radicals Name: Date: Simplifing rational epressions often requires the use of the rules for eponents and radicals. Simplifing rational epressions with radicals generall includes rationalizing the denominator. Rationalize the denominator when there is an irrational number in the denominator. Problem Set: Simplif the epressions. Rationalize the denominator if necessar.. 8 a b 7 6a b 6 7 6a b 7 6a b a b 7 6a b / a a 8 / b b a / / 6 / / / / 6 6 / / 6 6 / / / / / / 6 / / /. 7 7 / PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafch_k. of

6 Course Numbers and Their Properties. + + ( + ) ( + ) 6. u v 8u v 7 6 v 6u 6 6 u v ( 8) u u v v z 6 z 6 6 z z / / / / / / / 6 / z / / 6 z z / / / 9. / / / 0. 9 z 9 6 z z z z 6 z z 9 z 8z Reflection: Eplain in our own words what rationalizing the denominator involves. Answers will var. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafch_k. of

7 Course Numbers and Their Properties Module: Scientific Notation KEY Objective: To practice representing values using scientific notation Name: Date: Fill in the blanks. In scientific notation, numbers are represented as the product of a number and a power of 0. The number 600,000 is written in scientific notation as 6 0 /6 0. In scientific notation, less than 0. m 0 n, m must be greater than or equal to and When dividing numbers written in scientific notation, appl the quotient rule for eponents. That means the eponents are subtracted. Problem Set: Epress the following numbers in scientific notation , ,000, Translate the following numbers from scientific notation ,60,000,000,000, ,000,000, PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafdh_k. of

8 Course Numbers and Their Properties Indicate whether the following numbers are represented in proper scientific notation b writing correct or incorrect after each number incorrect correct correct incorrect Solve these problems. Epress ou answer in scientific notation.. (.7 0 ) + (. 0 ) (9.0 0 ) (. 0 t ) (9.6 0 ) (8. 0 ). 0 Etension Activit: Pretend ou are an algebra instructor teaching a lesson on adding, subtracting, multipling, or dividing numbers in scientific notation. Make up a stor problem that uses one of the above operations. Using another sheet of paper, write a lesson that shows our students how to solve this problem. Reflection: Name some professions that use scientific notation to represent numbers. phsicist, epidemiologist, biologist, mathematician, statistician How does scientific notation help people in these jobs? It makes calculations manageable and also provides a standard so that numbers can be quickl identified. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafdh_k. of

9 Course Numbers and Their Properties Module: Simplifing Algebraic Epressions KEY Objective: To practice simplifing algebraic epressions b collecting like terms and following grouping smbols Name: Date: Circle the like terms in this epression. z + z z + z z Describe what makes the others unlike terms. _different variables or the same variables to different powers. Fill in the blank. When ou collect and combine like terms, what part(s) of the terms are changed? coefficient (coefficient, variables, eponents) what part(s) of the terms are unchanged?_variables, eponents (coefficient, variables, eponents) what kind of change occurs?_addition or subtraction (addition or subtraction, multiplication) When removing an epression from parentheses or other grouping smbol, it's important to note the + or sign in front of the grouping smbol, as well as in front of the term inside the group. In the eample below, write the operator (+ / ) that belongs in the blanks. c (a + a b) c a a + b Problem Set: Simplif the epressions.. a ab + a b + ab b a + ab + a b b PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafeh_k. of

10 Course Numbers and Their Properties. 8b b b z z 8b + b z. m mn + m n + mn n 6. a + a a + a + 7a m mn + m n n a + a 0 7. b [-b (b b)] [ + (b )] 8. 6 {- + [ ( + )]} b (-b b + b) ( + b ) 6 (- + ( + ) b + b + b b b + 6 (- + + ) b b + 6 (- ) a [6a (7a + a)] [ (-a + 7)] 0. + {- [ + ( )]} -8a (6a 7a a) ( + a + 7) + (- ( + ) -8a (a 7a ) (a ) + (- ( ) -8a a + 7a a + + (- + ) -a 7a Reflection: Describe to a friend or classmate how to keep track of the positive and negative signs as ou simplif algebraic epressions. Include an tricks ou ve found helpful when this is especiall difficult to do correctl. Answers will var. Eample: I rewrite each line as I undo parentheses. I think to mself minus negative +. I double check each step. List to properties of numbers that ou can remember and find eamples of when ou use each one as ou simplif an algebraic epression. Answers will var. Eample: Distributive_ (-a + 7) (-a) 7 + a 7 Associative a 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafeh_k. of

11 Course Numbers and Their Properties KEY Module: Objective: Multipling Algebraic Epressions To practice multipling algebraic epressions, including a shortcut approach to multipling binomials Name: Date: Fill in the blanks. To multipl binomials, such as m mn o 8, do the following: multipl the _constants (or coefficients) E: 6 multipl the variables b adding the eponents. E: m mn o 8 m n o 8 Final answer: m mn o 8 6m n o 8 To multipl a monomial times a binomial, use the distributive propert and multipl the monomial b each term of the binomial. Eample: -( + ) (-) + _(-) To multipl a trinomial times a trinomial, multipl each term of one trinomial b other trinomial. Eample: (a + b c)(a b + c) a (a b + c) + b (a b + c) + -c (a b + c) A binomial squared follows this pattern: (a + b) a + ab + b st term middle term last term The sum and difference of two terms follows this pattern: (a + b)(a b) a + ab ab 0 + -b st term middle term last term a b simplified form To multipl this tpe of a binomial times a binomial, follow this pattern: ( + )( + ) st term middle term last term (Fill in the boes.) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maffh_k. of

12 Course Numbers and Their Properties Problem Set: Find the products.. a a b c a b c -. ( 8). - ( ) (a b + c)(a b c) 6. ( )( + ) a 6ab + ac + b + c + 7. (a + )(a ) 8. ( + ) a a ( ) 0. ( )( + ) Etension: Write down two sample patterns from the beginning of this activit, in equation form, and determine a value for each variable. Substitute those values into both sides of the problem (the equation) and verif that it comes out to a true statement. Eample: (a + b) a + ab + b, when a, b ( + ) Answers will var. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maffh_k. of

13 Course Numbers and Their Properties Module: Factoring Algebraic Epressions KEY Objective: To practice factoring algebraic epressions Name: Date: Fill in the blanks with one of the words in parentheses. Some polnomials can be factored completel b finding the greatest common factor: ac + ac c c(6ac + a ) (lowest, greatest) + (a + b) + ab ( a)( b) This pattern shows how to factor another tpe of polnomial. The sum of a and b must equal the coefficient of the middle term, while the product of a and b must equal the last term. (difference, sum, product, quotient) If the sign of the third term is positive, the signs of the factors a and b will be the same. (opposites, the same) In this case, the sign of the middle term determines whether both factors are positive or negative. (first, middle, third) The first two steps in factoring polnomials in the form a + b + c are ) to find the product (factors, product, sum) of a and c and ) to find factors of this value whose sum equals b. (a, b, c) In the final step, use the distributive propert. (associative, distributive) Problem Set: Look carefull at each polnomial to determine its form, then factor it.. 7a + a + 7a c d + 6cd 7a( + + ) ( 0)( + ) cd(c + d) 7a( + )( + ) ( + ) ( + 8)( + ) ( + 7)( + ) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part Document mafgh_k. of

14 Course Numbers and Their Properties g g 0 ( + )( + 0) ( 0.)(8 ) (g )(g + ) 0. c c (c 0)(c ) ( )( ) ( + ) ( )( ). 7c + c k k. g g (c + 0.)(6c + ) (k + )(7k ) (g 8)(g + ) Etension Activit: Here are some trinomials. Plug the values of a, b and c into the epression below to see if each one is factorable. b ac If the resulting value is an integer, the trinomial is factorable. Trinomial a + b + c g 9g Factorable? Value of b ac YES or NO ( 9) 7 8 no Factored form (if possible) + m 7m ( 7) ( ) 9 7 es ( )( + ) es (m )(m ) no Reflection: The epression above comes from the quadratic formula: b ± b ac a The term under the radical sign is called the discriminant. It determines the number and tpe of solutions a quadratic equation will have. What must be true about the value of the discriminant for a quadratic equation to be factorable? The discriminant must be a perfect square for the quadratic equation to be factorable. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part Document mafgh_k. of

15 Course Numbers and Their Properties Module: Factoring Sums and Differences of Perfect Cubes KEY Objective: To practice factoring the sums and differences of cubes Name: Date: The sum of cubes can be used to factor epressions such as Write the formula ou would use to factor this binomial as the product of a binomial and a trinomial. a + b ( a + b)( a ab + b ) The difference of cubes can be used to factor epressions such as Write the formula ou would use to factor this binomial as the product of a binomial and a trinomial. a b ( a b)( a + ab + b ) Sometimes ou must factor out a common factor before ou can appl the formula for perfect cubes. Problem Set: Factor these binomials as the product of a binomial and a trinomial ( ) + ( ) ( + )(9 + ) 9. m + 8n ( m ) + (n ) ( m + n )( m m n + 8 n 6 ) 9. ) ( ( )( ) 6. 6a + b ) ( a + b (a + b)(6a a b + b ) f + 6g ( f ) + (6g ) (f + 6g )(9f 8f g + 6g ) ( ) ( ) ( )( ) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafhh_k. of

16 Course Numbers and Their Properties 7. 6 z ( 6 ) ( z ) (6 z )(6 + 6 z + 8 z 0 ) 8. b + c b + (c ) ( b + c )( b bc + c 8 ) Factor out a common factor and then appl one of the formulas for perfect cubes np nr 9 n(7p r 9 ) n (p r )(9p + p r + r 6 6 ) (7 + 9 ) ( + )( ). 6 6w + 8w 6w ( + 8 6w ( + )( + 6 ) ) 9. 8 (8 7 9 ) ( )( ) 9. 8ab + 9ac 9 a (7b + 6c ) 0 a (b + c )(9b b c + 6c 6 ). 9 m n m p 9 m (8n p ) m (n p )(n + n p + 6 p 0 Reflection: It can be trick to keep the formulas for perfect cubes straight. How do ou remember the formula for the sum of perfect cubes and the difference of perfect cubes? Use an eample from above to demonstrate our strateg or thought process. Answers will var. The binomial in the sum of cubes is a sum and the trinomial has one subtraction sign, whereas the binomial in the difference of cubes is a difference and the trinomial has onl addition. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafhh_k. of

19 Course Numbers and Their Properties KEY Module: Objective: Rational Epressions: Simplif To practice simplifing rational epressions b factoring Name: Date: Complete each statement. A rational epression is the quotient of two polnomials. A rational epression is in simplest form if the numerator and denominator have as their onl common factor. To put a rational epression in simplest form, follow these steps:. Factor the numerator and denominator.. Cancel, or divide out, common factors. Removing negative signs or changing signs of factors can also help simplif rational epressions. Problem Set: Simplif each rational epression completel.. t 6 t + ( t )( t+ ) t + t ( + 7)( + ) + ( + 7)( ) ( + ) ( ) ( ) ( ) b. b b b b b b + ( )( ) 6. ( ) ( )( ) PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafjh_k. of

20 Course Numbers and Their Properties ab ab 9 a b b b a b b ( )( + ) + 0+ ( )( ) ( )( ) + ( + ) ( + + 6) ( + )( ) ( + )( + ) + Determine if each rational epression is in simplest form. If it is not, then write it in simplest form es. no, no,. + es Etension Activit: Anne had an algebra test esterda. One question was: Simplif the following epression completel and give reasons for each of our steps: 6 6 Determine if Anne s solution is correct and if her answer is simplified completel. Eplain our findings on a separate piece of paper. Anne s solution: 6 6 ( ) 6( ) factor out a common factor 6 6 ( )( + ) ( )( + ) ( )( + ) ( )( + ) factor the difference of squares cancel common factors reduce fraction to simplest form 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafjh_k. of

21 Course Numbers and Their Properties Student solutions should address these major points: Anne s first and second steps are correct. When she canceled common factors, she was correct to cancel the ( + ) and the ( + ) terms because the are equivalent. However, to cancel ( ) and ( ) is not correct. It is necessar to first factor - out of one of the terms. For eample, if - is factored out of ( ), then ( ) -(- + ) -( ). Now, the two ( ) terms can be canceled. And the problem can be completel simplified. The completel simplified answer to this test question is. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mafjh_k. of

24 Course Numbers and Their Properties KEY Module: Objective: Rational Epressions: Multipl and Divide To practice multipling and dividing rational epressions Name: Date: Read each statement about multipling and dividing. Decide whether it is true or false. When solving a multiplication or division problem with several rational epressions, use the order of operations. That is, multipl first, then divide. false (true, false) An answer is in simplified form if the numerator and denominator do not have an common factors other than. true (true, false) A simple fraction has one fraction bar. true true, false) A comple fraction must have a fractional numerator and a fractional denominator. false (true, false) Problem Set: Epress each product in simplest form. _ ab c a b c a b c. r r + 8 r + r PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maflh_k. of

25 Course Numbers and Their Properties Epress each quotient in simplest form a + a + a a. + ( ). b b + b + b b ( b + ) Reflection: Eplain how dividing rational epressions is related to multipling rational epressions. Be sure to tell wh ou must invert the divisor. The answers will var, but it should include the idea that division is the inverse of multiplication. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maflh_k. of

26 Course 6 Special Equations and Inequalities KEY Module: Objective: Evaluating Epressions with Absolute Value To practice evaluating simple equations that involve absolute value or distance Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. - if < 0 if 0 The absolute value smbol is treated like parentheses with regard to the order of operations Absolute value is computed from the inner most smbol to the outermost smbol. is the distance of from 0 or the origin on a number line. A measurement of distance must be positive. (negative, inverted, positive, even, odd) If ou are at - on a number line, then ou are units from zero. The distance between an two points a and b on a number line is given b the algebraic epression a 6. Problem Set: Evaluate the epression PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magah_k. of

27 Course 6 Special Equations and Inequalities Find 8 if Find - + if Find - + if Find if 7. Each pair of points lies on a number line. Find the unit distance between the two points.. -,9., ,-8. -,. -, 6. 0,8 8 Etension Activit: Below is a map of interstate highwa 8 stretching from Ligurta to Gila Bend. The distance markers are given in kilometers. The map is not to scale. Use the map to fill in the blanks that follow. A. Find the distance from Aztec to Ligurta in kilometers. B. Find the distance from Wellton to Sentinel in kilometers. 6 C. The equation -9 9 refers to the distance between Aztec and Tacuna. D. The equation 9 refers to the distance between Sentinel and Theba. E. The two cities that lie the same distance from Aztec are Wellton and Gila Bend. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magah_k. of

28 Course 6 Special Equations and Inequalities KEY Module: Objective: Absolute Value, Inequalities, and Interval Notation To practice using interval notation to describe solution sets for equations with absolute values and inequalities Name: Date: Fill in the blanks. A(n) open interval is a set of real numbers that contains neither endpoint but still contains all points in between. A(n) half-open interval is a set of real numbers that includes onl one endpoint. A(n) unbounded interval is a set of real numbers that approaches infinit. A(n) closed interval is a set of real numbers that contains both endpoints and all points in between. < if and onl if - < <. -9 or 9 if and onl if 9. For an real numbers a and b on a number line, the midpoint between (b - a) a and b is defined b this algebraic epression: a +. The distance from the midpoint of interval [a,b] to either endpoint is a-b defined b this algebraic epression:. Problem Set: Identif the interval as open, closed, half-open, or unbounded.. [,9) half open. (-,0) open 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magbh_k. of

29 Course 6 Special Equations and Inequalities. < unbounded. - < half open. - < < unbounded 6. [6,7] closed Graph the following intervals on the number line. You will need to mark our own points as needed. 7. (, ) { < } 8 Find the midpoint of the interval. Some answers ma be in the form of a decimal or fraction. 0. (-,9). [,) 7.. [-6,0] -. - < < Epress the following inequalities in interval notation.. { } [-,9]. { < } (-,) 6. { < } (-,6) 7. { + 7 } [-9,-] Reflection: Unbounded intervals approach infinit in either the positive direction, the negative direction, or both. Define infinit ( ) using our own words. Answers will var, but references to something that forever decreases or increases is the main idea. Can infinit ( ) ever be reached? Circle one. Yes No 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magbh_k. of

30 Course 6 Special Equations and Inequalities KEY Module: Objective: Graphing Linear Inequalities in Variable To practice graphing the solution sets to inequalities in variable Name: Date: Fill in the blanks. If ou multipl or divide both sides of an inequalit b a negative number, the inequalit sign will switch direction. A(n) union of two or more inequalities means that the solutions are graphed together on a single number line. A(n) intersection of two or more inequalities means that onl solutions common to all the inequalities are graphed. When solving inequalities, the word or is used interchangeabl with the smbol U. When solving inequalities, the word and is used interchangeabl with the smbol I. If the union or intersection of two or more inequalities ields no solution, then we sa that the solution set is a(n) empt set. Problem Set: Solve the inequalit for. Epress our answer as an inequalit <. + <. 9 + > < 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magch_k. of

31 Course 6 Special Equations and Inequalities Solve for all real numbers. Graph the solution set on the number line. You will need to mark our own points as needed. Some problems ma have no solution.. > or < I < - I < < 9 U < and 6 > 6 < - I 6 > < U < > 6 I or No solutions (empt set) All real numbers are solutions Etension Activit: You are studing for our final eam of the semester. Up to this point, ou have received eam scores of 98%, 8%, 8%, and 00%. To receive a grade of A in the class, ou need an average eam score between 90% and 00% for all eams including the final. Find the range of scores that ou can get on the final eam to receive a grade of A in the class. Write our answer as an inequalit final > 00 (.9) 0 00 final > 8 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magch_k. of

32 Course 6 Special Equations and Inequalities KEY Module: Objective: Graphing with Restrictions on the Variable To practice graphing the solution sets to absolute value inequalities in one variable Name: Date: Fill in the blanks. > 7 if and onl if < -7 or > if and onl if 9. The solution set to < - is a(n) empt set. The solution set to > - is all real numbers. < 6 if and onl if -6 < < 6. + > 8 if and onl if + < -8 or + > 8. Problem Set: Solve for. Write our answer as an inequalit.. > > -8 > -6 0 all real numbers or < < all real numbers empt set -6 < < 8 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magdh_k. of

33 Course 6 Special Equations and Inequalities Solve for. Graph the solution set on the number line. You will need to mark our own points as needed. Some problems ma have no solution > > or < < all real numbers - < < < -8 6 or -6 no solution (empt set) > + - > or < 6 Reflection: Eplain in our own words wh the solution to < -b is an empt set. When b is positive, the solution is an empt set because the absolute value on the left will alwas be positive, thus it will never be negative or smaller than a negative number. Eplain in our own words wh the solution to > -b is all real numbers. When b is positive, the solution is all real number because the absolute value on the left will alwas be positive. Therefore it will alwas be greater than a negative number. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document magdh_k. of

34 Course 6 Special Equations and Inequalities Module: Graphing Solution Sets of Associated Inequalities KEY Objective: To practice graphing the solution sets to quadratic inequalities in one variable and to other unions of solution sets Name: Date: Fill in the blanks. An inequalit of the form a + b + c < 0 is called a quadratic inequalit. For all real numbers a and b, a b < 0 if a < 0 and b > 0 or a > 0 and b < 0. For all real numbers a and b, a b > 0 if a > 0 and b > 0 or a < 0 and b < 0. The intersection of - and is. The union of - and is -. The intersection of < and > 9 ields a(n) empt set. If ou multipl or divide both sides of an inequalit b in a problem, ou must split the problem into two parts. For part, assume that > 0. For part, assume that < 0. Problem Set: Solving a quadratic inequalit requires four basic steps. Write the correct number of the step in the blank provided. Step Step Step Step Factor the left side of the inequalit. Write the inequalit in standard form. Check our sign assumptions b solving for the variable. Assume a sign, positive or negative, for each factor in the pair. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mageh_k. of

35 Course 6 Special Equations and Inequalities Solve for all real numbers. Graph the solution set on the number line. You will need to mark our own points as needed.. > ( )( + ) > 0 ( + 6) 0 > 0 and + > 0 or < 0 and + < 0 0 and or 0 and > and > - or < and < - 0 and -6 or 0 and -6 > or < - 0 or > > 0 ( )( ) 0 ( + )( + ) > 0 0 and 0 or 0 and 0 + > 0 or + < 0 and or and > - or < - or no solution -. + < > + < -9 > < 0 + > 0 ( + )( + 7) < 0 ( + 8)( ) > 0 if 0 + < 0 and + 7 > 0 or + > 0 and + 7 < 0 if > > 0 and > 0 or + 8 < 0 and < 0 < - and > -7 or > - and < -7 > -8 and > or < -8 and < Since > 0 no solution > or < -8 Since > 0 > if < 0 + > 0 and + 7 > 0 or + < 0 and + 7 < 0 if < 0 > - and > -7 or < - and < < 0 and > 0 or + 8 > 0 and < 0 > - or < -7 < -8 and > or > -8 and < Since < 0 < -7 U < < 0-8 < < Since < 0-8 < < Etension Activit: Eplain in words or numbers wh the quadratic inequalit + + < 0 has no solution (empt set). ( + )( + ) < 0; + < 0 and + > 0; < - and > - no solution 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mageh_k. of

36 Course 7 Coordinates and Curves Module: Calculating the Slope of a Line KEY Objective: To practice calculating the slope of a line from a graph or from two points Name: Date: Fill in the blanks. The slope of a line can be determined b finding the rise over the run. Lines that move from upper left to lower right have a negative slope. Lines that move from lower left to upper right have a positive slope. A line with a slope of zero is horizontal. A line with an undefined slope is vertical. Problem Set: Determine the slope of the line that contains each pair of points.. (,) and (0,). (-,-) and (,8). (,-) and (-,-7) m m m 7 6. (,6) and (,-0). (0,-8) and (-,-8) 6. (7, ) and (,) m no slope m m PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahah_k. of

37 Course 7 Coordinates and Curves 7 7. (-.,.) and (.7,-.6) 8. (,.8) and (6.,0) 9., 8 and, m m m Use the graph to find two additional points that lie on the line that has the given point and the given slope. 0. (,); m. (-,-); m (,) (-,) (-,) (-,0) (,-) (8,0) Etension Activit: The points (,), (6,0), and (k,) all lie on the same line. Find k. 0 m 6 Reflection: so... 0 k 6 k 6 k 6 7 so k Using the idea that slope rise, eplain wh vertical lines have no slope. run An vertical line will have a rise that is represented b some quantit. The run, however, will alwas be zero. Since an number divided b zero is undefined, vertical lines have no slope. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahah_k. of

38 Course 7 Coordinates and Curves Module: Point-Slope and Slope-Intercept Forms of Equations KEY Objective: To practice how to write the point-slope and the slope-intercept forms of linear equations Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. Given the slope m of a line and a point (a,b) on the line, the point-slope (point-slope, slope-intercept) form of the equation of a line is b m( a). The slope-intercept (point-slope, slope-intercept) form of the equation of a line is m + b where m is the slope and b is the -intercept. All vertical (horizontal, vertical) lines are defined b equations in the form a where a is a real number. Problem Set: On a separate piece of paper, write the equation of the line in point-slope form that satisfies the given conditions.. The line passes through (,-) and has a slope of ( ). The line passes through (-,-8) and has a slope of. + 8 ( + ). The line passes through (6,) and has a slope of -. - ( 6). The line passes through (-,-7) and has slope of ( + ) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahbh_k. of

39 Course 7 Coordinates and Curves On a separate piece of paper, write the equation of the line in slopeintercept form that satisfies the given conditions.. The line passes through the points (,-8) and (-,). -. The line passes through the points (,) and (7,). +. The line passes through the points (-9,-) and (-,6) The line passes through the points (-,-) and (0,8) Determine if the equation is true or false. On a separate piece of paper, show work to support each answer.. True False The slope of a line through the points (,) and (6,) is. The slope of a line through the points (,) and (6,) is.. True False A line represented b the equation - is a horizontal line. A line represented b the equation - is a vertical line.. True False A line with slope of - that passes through (-,) has this equation written in point-slope form: + -( ). A line with slope of - that passes through (-,) has this equation written in point-slope form: -( + ).. True False An equation in point-slope form can be rewritten in slopeintercept form. Eamples will var.. True False It is possible to determine the equation of a line in slopeintercept form simpl b knowing two points that are on the line. Eamples will var. Etension Activit: Triangle ABC has vertices at A (7,6), B (,), and C (7,-). Triangle EFG has vertices at E (,), F (-,-), and G (,-8). Sketch each triangle. Write the slope-intercept form for the equation of the line that represents each side of each triangle. With a partner, state to observations about the information ou have compiled. (Hint: You ma want to discuss the positions of the triangles, the slopes of the sides of each triangle, the size of the triangles, the lengths of the sides of the triangles, etc.) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahbh_k. of

40 Course 7 Coordinates and Curves A B E F C G AB: BC: AC: 7 EF: 6 FG: - 6 EG: Observations: (will var) ABC and EFG are the same size EFG is ABC shifted left units and down units. Each triangle has one side which is vertical, one side with a slope of 6, and one side with a slope of - 6. AB EF, BC FG, AC EG. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahbh_k. of

41 Course 7 Coordinates and Curves Module: Equation of a Line Given a Point and Parallel Line KEY Objective: Finding the equation of a line through a point and parallel to a given line Name: Date: Fill in the blanks. If two lines are parallel, the will have the same slope. If the equation of a line is given in the form m + b, ou can quickl determine the slope and the -intercept. The equation ( ) is in point-slope form. Name a point that ( ) (other possible answers). All lines parallel to the line ( ) passes through. _(, ) will have a slope of. Problem Set: Determine the equation that satisfies each of the following.. passes through (,) and is. passes through (-,) and is parallel to +. parallel to + 7 ( + ). + ( ) or ( + ) or + 6. passes through (-,-6) and is. passes through (-6,0) and is parallel to + 0. parallel to +. + ( 6) 0 + or ( + ) or + 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahch_k. of

42 Course 7 Coordinates and Curves. passes through (0,) and is 6. passes through (,6) and is parallel to +. parallel to + ( ). +, ( 0), 6 ( ) or + or + 7. passes through (,) and is 8. passes through (-,) and is parallel to 7. parallel to +. Etension Activit: Are an of the following lines parallel? If so, name those that are parallel. Eplain ( + 6) Eplanation: All three lines are parallel because the each have a slope of lines are distinct.. All three Reflection: If two lines have different slopes, describe at least one thing that is certain to be true. If two lines have different slopes, the will certainl intersect at eactl one point. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahch_k. of

43 Course 7 Coordinates and Curves Module: Equation of a Line Given a Point and Perpendicular Line KEY Objective: To practice finding the equation of a line through a point and parallel to a given line Name: Date: Fill in the blanks. Two lines are perpendicular if the7 intersect and form a right angle. If two lines are perpendicular, the product of their respective slopes is equal to -. Another wa of determining if two lines are perpendicular is to chec if one slope is the negative reciprocal of the other. The negative reciprocal of b a is b a. If a line has a slope of., a line perpendicular to it will have a slope of Problem Set: Determine an equation that satisfies each of the following.. Find the equation of a line. Find the equation of a line that that contains (, 7) and is that contains (-, ) and is perpendicular to. perpendicular to ( ) or + ( + ) or PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahdh_k. of

44 Course 7 Coordinates and Curves. Find the equation of a line. Find the equation of a line that contains (, ) and is passes through (-, -) and is perpendicular to 8 ( ). perpendicular to ( ) or 0 + ( + ) or. Find the equation of a line 6. Find the equation of the that passes through (, -6) line that passes through and is perpendicular to (-, -7) and is perpendicular 7. to +. (Think) ( ) or Determine if the following pair of equations are parallel, perpendicular, or neither ; 8. + ; perpendicular + 6 ( + ) parallel 9. 8 ; neither (same line) ; + ( ) neither Etension Activit: Find the value of k so that the line passing through (-,) and (,k) is perpendicular to +. An line perpendicular to will have a slope of. So: k ( ) or k 8 k k 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahdh_k. of

45 Course 7 Coordinates and Curves Module: Perpendicular Bisector of a Line Segment KEY Objective: To practice finding the equation of the perpendicular bisector of a line segment Name: Date: Fill in the blanks. If two lines are perpendicular, their slopes will be negative reciprocals. A perpendicular bisector of a line segment is not onl perpendicular to the segment but passes through the midpoint of the segment. To find the midpoint of a segment whose endpoints are (, ), and (, ), the -coordinate of the midpoint is found b taking + and the -coordinate of the midpoint is found b taking +. After finding a segment s midpoint and slope,. it is usuall easiest to put the equation of the perpendicular bisector in point-slope form. It is possible to find the perpendicular bisector of an line just as ou can find the perpendicular bisector of line segments. (true, false) 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maheh_k. of

46 Course 7 Coordinates and Curves Problem Set: Find the slope of the line segment with the given endpoints. Then give the slope of an line perpendicular to the segment.. (,-6) and (,7). (-8,) and (6,-) 7 ( 6) ( m ) ( m ) 6 6 ( 8) 7 ( m ) ( m ). (,) and (,-9). (-,0) and (,-) 9 ( m ) ( m ) 0 ( ) 6 6 ( m ) ( m ) Determine the midpoint of the following line segments with the given endpoints.. (-8,0) and (-,) 6. (-,-6) and (,) 8 + ( ), (,6), 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document maheh_k. of 7 ( 0, ) 7. (-6,-8) and (-,-) 8. (0,) and (,) 6 + ( ) 8 + ( ) 0 + +, (, ),, ( 6.,7.) Determine the equation of the perpendicular bisector of the line segment with the following endpoints. 9. (,) and (8,0) 0. (-,-) and (6,8) 7 ( ) ( 6). (-,) and (,-). (,8) and (6,0) + ( ) ( ). (,7) and (-,-). (0,6) and (-8,0) ( ) ( + ). (,) and (,) 6. (-0,6) and (-,-) 6 7 ( 6) ( + 6)

47 Course 7 Coordinates and Curves Module: Distance between Points KEY Objective: To practice finding the distance between points Name: Date: Fill in the blanks. The distance formula is derived from the Pthagorean theorem, which is c a + b. It is often helpful when finding the distance between two points to create a right (left, right) triangle. Given two points (, ) and (, ), the distance between them is d ( ) + ( ) Problem Set: Find the distance between the following points. Round our answer to the nearest hundredth.. (,) and (,7). (-,) and (-7,0) d ( ) + ( 7 ) d ( 7 ) + ( 0 ) + ( ) (,-) and (,-). (-,-) and (-8,-) d ( ) + ( ) d ( 8 ) + ( ) ( ) + ( 9) ( 6) + ( ) PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahfh_k. of

48 Course 7 Coordinates and Curves. (-,) and (,-6) 6. (-,-) and (,9) d ( ) + ( 6 ) d ( ) + ( 9 ) + ( ) (0,6) and (,0) 8. (,) and (0,-) d ( 0) + ( 0 6) d ( 0 ) + ( ) + ( 6) ( ) + ( 7) (,8) and (-,8) 0. (-9,-) and (,7) d ( ) + ( 8 8) d ( 9) + ( 7 ) ( ) Etension Activit: The distance between (6,) and (,) is. Find. ( 6) + ( ) + ( ) + ( ) 0 ( ) 69 + ( ) ( ) so: or PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahfh_k. of

49 Course 7 Coordinates and Curves Module: Distance between a Point and a Line KEY Objective: To practice finding the distance between a line and a point not on the line Name: Date: Fill in the blanks. Use one of the words in parentheses when choices are given. The distance from a line to a point not on the line is alwas represented b the perpendicular (parallel, perpendicular) distance between them. If two lines are perpendicular, their slopes will be negative (positive, negative) reciprocals of each other. Given two equations of perpendicular lines, one can find the point of intersection b solving a sstem (sstem, group) of equations. Once the point of intersection is found for two perpendicular lines, use the distance formula (distance formula, distance rule) to find the distance between the original point and the point of intersection. The distance between two points is alwas positive. (negative, positive) Problem Set: Find the distance from the given point to the given line. Round our answer to the nearest hundredth.. (-,) and. (,) and + - ( ) ( + ) ( ) + - equation of - equation of + 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahgh_k. of

50 Course 7 + Coordinates and Curves Distance between (, ) and (,) Distance between (,) and (,0) d ( ) + ( ) d ( ) + ( 0 ) + ( ) ( ) + ( ) (-,) and + ( ). (6,) and - + ( + ) ( 6) 8- equation of + - equation of + + ( 8) ( + ) Distance between (-,) and (-,-) Distance between (6,) and (0,) d ( ) + ( ) d ( 0 6) + ( ) + ( 6) ( 6) + ( ) PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahgh_k. of

51 Course 7 Coordinates and Curves. (-,-) and + 6. (-7,-) and ( + ) + ( + 7) Distance between (-,-) and (0,) Distance between (-7,-) and (0,-) d ( 6 ) + ( ) d ( 0 7) + ( ) 7 + ( ) Reflection: When describing the distance between a line and a point not on the line, wh do we use the perpendicular distance? There are man distances between a point and a line. We need to make the assumption that it is the shortest distance that we want to find. The shortest distance will alwas be found along the perpendicular. 00 PLATO Learning, Inc. All rights reserved. Printed in the United States of America. PLATO is a registered trademark of PLATO Learning, Inc. Algebra, Part, Document mahgh_k. of

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