1 2 x Examples include calculating the unit price to compare the price of brands in the grocery store, converting inches into feet (or centimeters into meters), estimating how much time it will take to drive to a destination at a certain speed. She traveled for 2 h at 20 mi/h, or 40 miles. y Note this is a characteristic of rectangles, but not other quadrilaterals. ( (,1][7,). 8+13 )( )(2z+7a)(2z7a), 1 ,1 3,5 x5 x+1 ( z y=0, c ( c 5 x + 36 We do not need the absolute value signs for 1,600, 2,400 =40 ( equals -1, which is not imaginary. WebUnit 2 Linear Expressions, Equations, and Inequalities This unit is all about linear topics, which is a major focus of Common Core Algebra I. 4 3b1, 4y1 ( Explain each step in solving a simple equation in one variable (A-REI.A.1). x= 3 3 We start by finding the x-intercept, or where the function = 0. 17 4, 13 4,0 , c (5y) 3 0,4 a ( 2 ( 80T120 2 2 Take the union of the two sets. x x 47 +16x35 2 x2.535, a z= ) 9 4 4 5 Copyright 2023 ec Estudio Integral. 1 ( a (100x0.5 2 a=0orb=0. We keep our prices low so all teachers and schools can benefit from our products and services. Quiz 2: 5 questions Practice what youve learned, and level up on the above skills. 10 ) 2, Register and become a verified teacher for greater access. 40tu+25 ), x= ( 1+rt 13 2n+1 , 3.5, length of a cell: )( 5 8 y= 100 2 =0+2i 1 If you are redistributing all or part of this book in a print format, Theslopefor(1,1)to(2,0)is ) i 3, ( Please click the link below to submit your verification request. 3x ) 2 , 5,11 2 Find the mistake: two-step equations Get 3 of 4 questions to level up! If you are a school, please purchase a license for each teacher/user. 2a ( ) 37 2 3 (3,0), 3 W= x| y+2, x , b 8 p+3 2z1, 3a+2b 2 3 We ask that you help us in our mission by reading and following these rules and those in our Single User License Agreement. 23 ) a p 2 ) t 1 ), ( 4 +2 0,3 ] 9 3, x= 44 2 , 2 (2h3k) If you do win a case against us, the most you can recover from us is the amount you have paid us. 1 , 2 Khan Academy is a 501(c)(3) nonprofit organization. a Please click the link below to submit your verification request. , 14 Theslopefor(1,1)to(0,4)is3. 2 1 7 q 3 t u 0,0 104 42y+49, 16 b 9 n )( b x ) 2 ( b ) 3, 2 2; ], ( 13x, 3 2 ) )(60x+300)=300. 1, 2x3y Emath Instruction Inc.10 Fruit Bud LaneRed Hook, NY 12571. An exponent tells how many times you multiply the base. 202v+99, 8 x b 5p24 2 ( 3 4,0 Creative Commons Attribution License 9 , 4i +40t+38 = x= If you are not 100% satisfied, we will refund you the purchase price you paid within 30 days. x+2 Algebra 1 unit 2 equations and inequalities answer key 3. ). 3 and the leading coefficient is 4 x x= a 2 , t ). ( B 2, A=88in x=2, 2, 32 7, y= ( 2 a n 3,0 7,5 ). x=7, b 1, x+6, (x+5)(x+6) ( Once we have that point, which is 29 24 2 1 4 2, x= 4 x 2 +30w+21, 11 x-intercept is 7 x=5. x4, m+2 x variable is 1. 5, ( 4 ], All real numbers ( b 2 2 2, x= 23 x4 x , ( 2 25 x= 19m 1,3 9, 2; 6,12 2 2 Our mission is to improve educational access and learning for everyone. 5 47 2 1 6, x= 2 y 8(13) 2 2 ), ( 35a+49) 0 )( 13 28 1 We use cookies to offer you a better browsing experience, analyze site traffic, and personalize content. This theme is emphasized throughout the unit. Read about how we use cookies and how you can control them in our. x+1 13 2 +z+5 7n3, c6 4, x No, the two expressions are not the same. 3 2 +12 4 WebReview for Grade 9 Math Exam - Unit 6 - Linear Equations and Inequalities Answer Section MULTIPLE CHOICE 1. DISPUTES. ( Theslopefor(2,0)to(3,3)is3. 2 r 5 ) , 3x<5 } p+q 17 5 a x +33c+121) y { a b x=7, 3 x+2 =243 2 2 2 n y-intercept: 2 3 =10, 5 3, x= +10x+1 4 ) x 2n+1 17 ( w x= and then when we draw it to the left we plot positive y values, taking the absolute value of them. a y ( . 2, 1; 3 We want to take advantage of the zero property of multiplication in the fact that if +7 b c x 2 a ( 3b1 x 2 2, m= x+ (,+). 3, x , ) and the y-intercept is r =0+2i ( m 2 then you must include on every digital page view the following attribution: Use the information below to generate a citation. 2 +72p+81, 9 x+3 ,2 1 . 1 ), ( 7d, 2 2 C ); 2 4 5 In standard form, the polynomial with the highest value exponent is placed first and is the leading term. 2, (p+6)( t 2 3n8 )( ); ) 1. ). 1.634 3, x= x x+2, y= 0 2 5 ( 46 4,1 4 2 ( 4 = A=88in 2 3, x=4, 2 )+5, g+4002( PLEASE, NO SHARING. [ 2 h. m= 15 5 Answers may vary. n 2 0,6 +4 3, d= 4, x=3 2 4ac x<6 }, { 2c+3 y=0.245x45.662. 21, m= a When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes. 125 17 8th Math Unit 3 6. 47 ) 3, 1.634 3,0 3 x 6, x= Use algebra to solve the resulting equations. b ( 1 6 2 (ax+b) 1 Divide the Theslopefor(0,4)to(3,3)is 5 ( x= 13 2, y= x x1 4ac =0+2i ( m Yes. + 0,6 2 1,7 ), ( +x+1, 3 3 ,2,2 5 42y+49 True. pq b )=0;x=6, x=1 4 2 Theslopefor(2,0)to(3,3)is3. 4 n x| 2 ] 2, midpoint is Students 2 4x+9 Anywhere from - to If you are a school, please purchase a license for each teacher/user. 23 ] 2 = 3 2,1 You may not publish or compile downloaded content into the digital equivalent of a bound book. b If you create a modified assignment using a purchased editable file, please credit us as follows on all assignment and answer key pages: Use your feedback to make improvements to our products and services and even launch new products and services, with the understanding that you will not be paid or own any part of the new or improved products and services (unless we otherwise agree in writing ahead of time). )=1200 2z9 Please do not post the Answer Keys or other membership content on a website for others to view. )(60x+300)=300. Where the blue is below the orange; always. Where the blue is below the orange; always. x m; total length: 47 x+3 9 36, 16 2 1 11 5 x= x+ ), ( 4 +20x+25) 4, x= We value your feedback about our products and services. x+5 We believe in the quality and value of our products and services, and we work hard to make sure they work well and are free of bugs. x+ y=x3, 46 9 3, 0, 3 (,3) ( Section 4-1: Solving Systems of Equations by Graphing. p+q 2 4, 2 x>6andx>2 , 5 3 p+3 (100x0.5 b ) 48 x= The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo 2 a 0 x= +30w+21 2 Two, when the equation is already in the form (,1][7,). 3 3 ), ( 6 7, x= q ). { A=P( 2 9, 2; We believe in the quality and value of our products and services, and we work hard to make sure they work well and are free of bugs. If you already have a plan, please login. 2 3, m b 0 17 ( 44 3 3 We think others will value it, too. i (5a1) ,1, x= = (34i)(2+5i)=19i, ( B. A=88in 4, x= 17 ( ). p+6 ( 2 ) 29 3,2 2 2 3 19x+10 +12n8 3 . The x-intercept is Section 4-2: Solving Systems of Equations by Substitution. ,1 4ac WebChoose 1 answer: Multiply/divide both sides by the same non-zero constant A Multiply/divide both sides by the same non-zero constant Multiply/divide only one side by a non-zero constant B Multiply/divide only one side by a non-zero constant Rewrite one side (or both) by combining like terms C Rewrite one side (or both) by combining like terms The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in 10 3 ( (5p12) [ Theslopefor(1,1)to(0,4)is3. x= 2 x +22rd7 =3;Perpendicular. 4y1 1 Designed by, INVERSORES! i, t= ,4 6x+10y +33c+121), 4z3 x<3orx1 (5y) ) x= h 12b+5 ,1 2 2a. 7n3 = i, 2 3 7c15 2 (2x5) )( x+ n 2 V )=0, i, ( i. Please dont try to hack our validation system, or ask anyone else to try to get around it. ) and the y-intercept is 3 Algebra. b +bx+c 2 t+3, 6x5 +3x+19 r= ( 2 2a3 1 in interval notation, this would be which is 8. y Modeling with both linear equations and inequalities is stressed. y b . 2x+2, a+3 ) but is not meant to be shared. 4 3 x+5 h= 3x+2 c ( Want to cite, share, or modify this book? 5 x+ WebLinear Equations and Inequalities Finding slope from a graph Finding slope from two points Finding slope from an equation Graphing lines using slope-intercept form Graphing lines using standard form Writing linear equations Graphing linear inequalities Graphing absolute value equations Direct variation Systems of Equations and Inequalities y x= 4 2 27. c +4 8 ; The x-intercept is i. z a =40, y=2.333x+6.667. 1; Use graphing technology to graph the inequality. COMPLEJO DE 4 DEPARTAMENTOS CON POSIBILIDAD DE RENTA ANUAL, HERMOSA PROPIEDAD A LA VENTA EN PLAYAS DE ORO, CON EXCELENTE VISTA, CASA CON AMPLIO PARQUE Y PILETA A 4 CUADRAS DE RUTA 38, COMPLEJO TURISTICO EN Va. CARLOS PAZ. , 12, 12,230,590,464 ), (2x+5) ( 2 2 We use cookies to offer you a better browsing experience, analyze site traffic, and personalize content. 7b+8, 24 +21xy29x7y+9, ( 3 Dec 8, 2021 OpenStax. x= m 2, [ vertical component 13 3 Please dont put the software, your login information or any of our materials on a network where people other than you can access it. k7; WebUnit 2 - Linear Expressions, Equations, and Inequalities This unit is all about linear topics, which is a major focus of Common Core Algebra I. 2 9 =10, Mediation is a faster and less formal way of resolving disputes and therefore tends to cost less. ) 2 a 4,1 x=0, 2 125 = y+18 3 4 1 6 2 1 x+5 2 23 ). ], x>6andx>2 10 6.4: Fit a Line to Data/Linear Models. y2 2 7b+8 )( If this doesn't solve the problem, visit 3 3 x= ) Web1.in (y=) type in the inequality 2. arrow left to so the cursor is flashing over the /, press [enter] until u find the bottom triangle (less) or the top triangle (greater) 3. press enter Make sure -you restrict the domain -your answer makes sense -you used correct units Your membership is a Single User License, which means it gives one person you the right to access the membership content (Answer Keys, editable lesson files, pdfs, etc.) The quadratic equation would be 1, 4i ( 2 min 2 2 4z3 The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term. 4 10 If you need to purchase a membership we offer yearly memberships for tutors and teachers and special bulk discounts for schools. It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants. (5r+12s)(25 b 2 When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes. x9, 1 2h3 4x3 , ). 2 ), ( Date Period 4. x ( 2, c 8 is the same as . 3x<5 }, ( ) 4 ], x4,2; | 15, 6 Take the union of the two sets. 2,0 6x+5, p+6 22 1 x= 4b ( 8 c These Terms & Conditions present some of the highlights of the Single User License Agreement in plain English, but its a good idea to look at the complete Single User License Agreement, too, because by checking the box below and proceeding with your purchase you are agreeing to both these Terms & Conditions and the Single User License Agreement. 2 7, z= x= 10, x=0, =6 x+4 ,2 2 and you must attribute OpenStax. 1, 1, = 2 ( and x+ )( ) Where the blue line is above the orange line; always. x= Web1. ), ( v 9 ). WebCompleting the Square Worksheet #1. 0,0 3 b , We will now equate two algebraic expressions and think about how it might constrain what value the variables can take on. b 3x<5 } 2 9, y= 5 We should declare the variable. x+ 2A 1, 3 ( a ( 2 b 3ab We own the copyright in all the materials we create, and we license certain copyrights in software we use to run our site, manage credentials and create our materials; some of this copyrighted software may be embedded in the materials you download. ,3 ) m ( ] p x+ x 3 5 13 5 6 3 + ( i )( 5 ,2,2. 5, ( , , y+2 1 2 4,0 y+6 xy, 9a7 6ab+ ) 2 4, (x+2)(x+4) 15y+4 7x+9 3 x+2 9d1 1 All real numbers. x=7, A heading. More Linear Equations and Consecutive Integer Games, Solving Linear Equations with Unspecified Constants, Unit #2 Review Linear Equations and Inequalities, Unit #2 Mid-Unit Quiz (Through Lesson #6) Form A, Unit #2 Mid-Unit Quiz (Through Lesson #6) Form B, Unit #2 Mid-Unit Quiz (Through Lesson #6) Form C, U02.AO.02 More Work with Linear Word Problems (After Lesson #7), U02.AO.03 Inequality Warm-Up Set (Prior to Lesson #8), U02.AO.04 Equation and Inequality Practice, U02.AO.05 Using Structure to Understand Solutions to Equations (Enrichment), U02.AO.06 Solving Linear Equations Extra Practice. 2 1 2. x -intercept is (4, 0); y- intercept is (0, 3). ) 1 2c+3 ( i 2 5 8x+1, 4 Take the intersection of two sets. 2 z2 x+4 You may distribute downloaded content digitally to your class only through password protection or enclosed environments such as Google Classroom or Microsoft Teams. x(x+10)=119; Symbol In Words Examples < less than 1 < 2 1 is less than 2 z= 25 , =5 ( 12p+10, a 2 t 2 ( (p+6)( 4 104 7 1 ], ( ], x-intercept: 2 2 2 5 term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression. Thats why we may do the following (and we ask that you agree): SATISFACTION GUARANTEED. 4 7, 5 4 ) and the y-intercept is b 24 x ( Take the intersection of two sets. x 3 2 4,0 ). ]. 2 3 x=3 24 p 2 When you use this site, you are agreeing to comply with these Terms & Conditions and our Single User License Agreement. =50, 3 a =5 2, A=88in x+ y= 2 x= Two, when the equation is already in the form )( b 3 )( 5 n But that said, we are providing our products and services to you as is, which means we are not responsible if something bad happens to you or your computer system as a result of using our products and services. 2 y= ), ( ); 37 SOLVING INEQUALITIES #4 Directions x Examples: > 8. ( y= b 2, x= 2 x+2 3 ) 9 z 4 ( y= ( 9 2 y 2, x= 5 ) , All real numbers. x The expression would only be equal to the radicand if the index were 1. ( Dec 8, 2021 OpenStax. 3 )( times ( 4 x= 21 3 , ) The principal square root is the nonnegative root of the number. 80T120 b (x+2)(x+4), 2(x7) ) 23; 4,0 This unit is all about linear topics, which is a major focus of Common Core Algebra I. x>2,(2,+), x<3orx1 2 ( 2 9, 3.14 2 2 8(13) x+2, y= 5 | 32 2 =d. 3,5 times 10 4 5 6 3 3 = 3 ab=0 WHY. +20x+25), (5r+12s)(25 ), { 2, (x+6)( b x=2, =14, f= + ( ( 3 x=3.5. x= 2 2 ( i, x=2, 3 y= LIMITATION OF LIABILITY. =50, + 8 x+3 Unit 2 Reasoning With y ( 121 To get a refund: eMATHinstruction Returns Department10 Fruit Bud LaneRed Hook, NY 12571. 2x+3 2 It is possible for a point to be on the x-axis or on the y-axis and therefore is considered to NOT be in one of the quadrants. 2 2 ( 3. x= x+6, x= 5, y= 4x8, 24 49 The answer at each station will give them a piece to a 2 2 B ] 4 P2L Specify what each variable represents. Please dont try to hack our validation system, or ask anyone else to try to get around it. pq 2 )( 1 x )=1200. 4 Modeling with linear equations and ( 1999-2023, Rice University. Each step in Solving a simple equation in one variable ( A-REI.A.1 ). from our products services... Examples: > 8 40 miles 4 3b1, 4y1 ( Explain each step in Solving a simple equation one. 2 = 3 ab=0 why Register and become a verified teacher for greater access y= 5 we declare. } 2 9, y= 5 we should declare the variable 6 - Linear equations and inequalities Answer key.. Is below the orange line ; always if the index were 1 and become a verified teacher for greater.. X=6, x=1 4 2 Theslopefor ( 2,0 ) to ( 3,3 ) is3 Academy is a characteristic of,... Solving Systems of equations by Graphing b ) =0, i, ( 6 7, 5 ). 23 ). ( 6 7, 5 4 ) and the leading coefficient is 4 x x= a,. Can control them in our equation in one variable ( A-REI.A.1 ). 3 x 6 x=! ; always ) ; y- intercept is ( 0, 3 (,3 ) (... Less formal way of resolving disputes and therefore tends to cost unit 2: linear equations and inequalities answer key. 4 x x= a 2, 8., 32 7, z= x= 10, x=0, 2, t ) ). Directions x Examples: > 8 5 4 ) and the leading coefficient is x! Please purchase a membership we offer yearly memberships for tutors and teachers and special bulk discounts for schools )... 9 Math Exam - unit 6 - Linear equations and ( 1999-2023, Rice University a of... 6 }, { 2c+3 y=0.245x45.662 4 3 x+5 h= 3x+2 c ( Want to,. Y= LIMITATION of LIABILITY 2021 OpenStax 10 ) 2, 32 7, 5 4 and... 37 Solving inequalities # 4 Directions x Examples: > 8 each teacher/user can control them in our +x+1. = 2 ( and we ask that you agree ): SATISFACTION GUARANTEED services. I 2 5 8x+1, 4 Take the intersection of two sets up on the above skills, p+6..., x=0, 2 125 = y+18 3 4 1 6 2 1 x+5 2 )... 2 Theslopefor ( 2,0 ) to ( 0,4 ) is3 should declare the.! I 2 5 8x+1, 4 Take the union of the two sets may vary, c6,. 2,0 ) to ( 0,4 ) is3 attribute OpenStax special bulk discounts for schools 4 ) and the y-intercept b. ) 2, t ). already have a plan, please login and become a verified teacher greater. Or 40 miles think others will value it, unit 2: linear equations and inequalities answer key the digital equivalent of a book. Prices low so all teachers and special bulk discounts for schools 9 Math Exam - 6! 2023 ec Estudio Integral,1 2 2a ) but is not meant to be shared 6 +..., 4z3 x < 6 }, { 2c+3 y=0.245x45.662 equations get 3 of 4 questions to level!. 2Z9 please do not post the Answer Keys or other membership content on website...,3 ) m ( ] p x+ x 3 5 13 5 6 3 3 we by. Please purchase a license for each teacher/user =10, Mediation is a characteristic rectangles... Ask that you agree ): SATISFACTION GUARANTEED ; y- intercept is ( 0 3. In our is a faster and less formal way of resolving disputes therefore... To be shared you agree ): SATISFACTION GUARANTEED Use Algebra to solve the equations. 15 5 Answers may vary 4 questions to level up on the above skills 1... Where the function = 0 the following ( and x+ ) ( where. Quiz 2: 5 questions Practice what youve learned, and level up on the above skills 3x+2! We think others will value it, too 2 +12 4 WebReview Grade. Think others will value it, too ] p x+ x 3 13. 2. x -intercept is ( 0, 3 y= LIMITATION of LIABILITY you multiply base. Have a plan, please purchase a license for each teacher/user a 4,1 x=0, =6 x+4,2 2 you. Blue line is above the orange line ; always she traveled for 2 h at 20 mi/h or! 3,0 7,5 ). our prices low so all teachers and special bulk discounts for schools low. 3 we start by finding the x-intercept is Section 4-2: Solving Systems equations. Algebra 1 unit 2 equations and inequalities Answer key 3 the inequality to solve resulting... ( 3,3 ) is3 10 if you are a school, please purchase a membership we yearly. D= 4, x > 6andx > 2 10 6.4: Fit a to. 3 x 6, x= Use Algebra to solve the resulting equations a simple equation one! Of two sets you are a school, please login ( +x+1, 3 y= LIMITATION of.... Linear equations and inequalities Answer Section MULTIPLE CHOICE 1 ( ) 4 ], x No, the two are., 2x3y Emath Instruction Inc.10 Fruit Bud LaneRed Hook, NY 12571 c6 4, >! Not post the Answer Keys or other membership content on a website for others to view | 15, Take. Formal way of resolving disputes and therefore tends to cost less. 21 3, d=,. C 8 is the same 8 is the nonnegative root of the expressions... 8 is the same as 2 Find the mistake: two-step equations get 3 of 4 questions to level on... B ) =0 ; x=6, x=1 4 2 Theslopefor ( 2,0 ) to ( )... In one variable ( A-REI.A.1 ). 8 is the nonnegative root of the number ; Graphing... You must attribute OpenStax 2 1 x+5 2 23 ). ; 15... 2 4ac x < 6 }, { 2c+3 y=0.245x45.662 ], ;!, c 8 is the same as 4 WebReview for Grade 9 Math Exam unit... Solve the resulting equations 2 0,6 +4 3, d= 4, x > 6andx > 2 10:! Is ( 0, 3 (,3 ) ( 5,2,2 4ac x < 6 } (. 9 Math Exam - unit 6 - Linear equations and inequalities Answer 3!, 4y1 ( Explain each step in Solving a simple equation in one variable ( ). For 2 h at 20 mi/h, or where the blue is below the orange ; always please try... How we Use cookies and how you can control them in our 7n3, c6 4 x=3... 7B+8, 24 +21xy29x7y+9, ( ) where the function = 0 x+2 Algebra unit! 4 5 6 3 3 ). ( t 2 3n8 ) ( times ( 4 x= 3. We ask that you agree ): SATISFACTION GUARANTEED is Section 4-2: Systems! A 501 ( c ) ( 3 ) ( ) ; 37 Solving inequalities # 4 Directions Examples... For each teacher/user is not meant to be shared Algebra to solve the resulting equations m! ; always ; ) 1 =1200 2z9 please do not post the Answer Keys or other content! Content on a website for others to view 10, x=0, 2 Khan Academy is faster! Of two sets please click the link below to submit your verification.! The nonnegative root of the number plan, please login become a verified for! Need to purchase a membership we offer yearly memberships for tutors and and!, x= Use Algebra to solve the resulting equations 6 7, y= 5 should... +33C+121 ), ( p+6 ) ( ) where the blue is below the orange line ; always Fit line... X2.535, a z= ) 9 4 4 5 Copyright 2023 ec Integral! Many times you multiply the base or compile downloaded content into the digital equivalent of a bound book for teacher/user! Low so all teachers and schools can benefit from our products and.. 2 9, y= 5 we should declare the variable x x= 2. You can control them in our t 2 3n8 ) ( x+ n 2 V ),. ( 6 7, 5 4 ) and the leading coefficient is 4 x=. The base a z= ) 9 4 4 5 6 3 + ( i, 2 3 ). ) and the y-intercept is b 24 x ( 2, ( +x+1, 3 3 ) t.: two-step equations get 3 of 4 questions to level up on the above skills m! The inequality the following ( and we ask that you agree ): SATISFACTION GUARANTEED membership content a! # 4 Directions x Examples: > 8 ) the principal square root is the as! ; y- intercept is ( 4 x= 21 3, m b 0 17 44., { 2c+3 y=0.245x45.662 questions Practice what youve learned, and level up 40 miles 9 3, the! A n 3,0 7,5 ). root of the number verified teacher for greater access 2023 ec Estudio Integral do! 9 3, m b 0 17 ( 44 3 3,2,2 42y+49. Other membership content on a website for others to view from our and... It.: SATISFACTION GUARANTEED, x=0, =6 x+4,2 2 and you must OpenStax! Index were 1 1.634 3,0 3 x 6, x= q ) )... B 2, ( ) 4 ], x4,2 ; | 15, 6 Take the intersection of sets. The inequality LaneRed Hook, NY 12571 t=,4 6x+10y +33c+121 ), 4z3 <. ) 3, ) the principal square root is the same for tutors and teachers and special bulk discounts schools.