Includes Online Access: ~ Computer Adaptive PractIce Exams ~ Bonus Question Bank for Equations, Inequalities, "VICs See page for details 9rtanhattan G MAT the new standard Learn using Superior Tools developed by Superior GMAT Instructors • Scored in 99th percentile on the GMAT • Selected by rigorous face-to-face audition •Trained 100+ hours before teaching • Paid up to 4x the industry standard you're SERIOUSabout getting a GREATSCOREon the GMAT; you have to go with Off MANHATTAN GMAT." - Student at top b-school The Manhattan GMAT Advantage: Sophisticated Strategies For Top SCi GMAT and GMAC are reqlstered trademarks of tile Gra.L2ce Management Admission Council which neither sponsors nor endor 9danliattanG MAT'Prep the new standard BASIC EQ.UATIONS 11 In Action Problems Solutions 23 25 EQ.UATIONS WITH EXPONENTS 29 In Action Problems Solutions 35 37 Q.UADRATIC EQ.UATIONS 41 In Action Problems Solutions 49 51 4.FO~ 55 In Action Problems Solutions 63 65 FUNCTIONS 69 In Action Problems Solutions 79 81 INEQ.UALITIES 83 In Action Problems Solutions 103 105 VICS 107 In Action Problems Solutions 123 125 STRATEGIES FOR DATA SUFFICIENCY 131 Sample Data Sufficiency Rephrasing 137 OFFICIAL GUIDE PROBLEMS: PART I 143 Problem Solving List Data Sufficiency List 146 147 PART!: GENERAL TABLE OF CONTENTS :ManliattanG MAT·Prep the new standard 10 EQUATIONS: ADVANCED 149 In Action Problems Solutions 157 159 11 FORMULAS & FUNCTIONS: ADVANCED 163 In Action Problems Solutions 12 INEQUAUTIES: 173 175 ADVANCED 179 In Action Problems Solutions 187 189 13 ADDITIONAL VIC PROBLEMS 193 In Action Problems Solutions 195 197 14 OFFICIAL GUIDE PROBLEMS: PART II 201 Problem Solving List Data Sufficiency List 204 205 PART II: ADVANCED TABLE OF CONTENTS Chapter -~'-'-o/- ··.·i.: EQUATIONS, INEQUALIn~, BASIC EQUATIONS " ' & VIes In This Chapter • Solving One-Variable Equations • Simultaneous Equations: Solving by Substitution • Simultaneous Equations: Solving by Combination • Simultaneous Equations: Three Equations • Mismatch Problems • Combo Problems: Manipulations • Testing Combos in Data Sufficiency • Absolute Value Equations BASIC EQUATIONS STRATEGY Chapter BASIC EQUATIONS Algebra is one of the major math topics tested on the GMAT Your ability to solve equations is an essential component of your success on the exam Basic GMAT equations are those that DO NOT involve exponents The GMAT expects you to solve several different types of BASIC equations: 1) 2) 3) 4) 5) An equation with variable Simultaneous equations with or variables Mismatch equations Combos Equations with absolute value To solve basic equations, remember that whatever Several of the preceding basic equation types probably look familiar to you Others-particularly Mismatch Equations and Combos-are unique GMAT favorites that run counter to some of the rules you may have learned in high-school algebra Becoming attuned to the particular subtleties of GMAT equations can be the difference between an average store and an excellent one Solving One-Variable Equations Equations with one variable should be familiar to you from previous encounters with algebra In order to solve one-variable equations, simply isolate the variable on one side of the equation In doing so, make sure you perform identical operations to both sides of the equation Here are three examples: 3x+ = 26 3x= 21 x=7 w= 17w-l 0= 16w- 1 = 16w Subtract from both sides Divide both sides by is the solution of the given equation Subtract w from both sides Add to both sides Divide both sides by 16 -=w 16 +3=5 Subtract from both sides Multiply both sides by 9danfiattanGMAIPrep the new standard you to one side you must also to the other side Chapter BASIC EQUATIONS STRATEGY Simultaneous Equations: Solving by Substitution Sometimes the GMAT asks you to solve a system of equations with more than one variable You might be given two equations with two variables, or perhaps three equations with three variables In either case, there are two primary ways of solving simultaneous equations: by substitution or by combination Solve the following for x and y x+y=9 2x= Sy+ Use substitution whenev- Solve the first equation for x At this point, you will not get a number, of course er one variable can be easily expressed in terms of the other x+y=9 x=9-y Substitute this expression involvingy into the second equation wherever x appears 2x= 5y+4 2(9 - y) = 5y + Solve the second equation for y You will now get a number for y 2(9 - y) = 5y + 18 - 2y= 5y+4 14 = 7y 2=y Substitute your solution for y into the first equation in order to solve for x x+y=9 x+2=9 x=7 :ManliattanG MAT·Prep the new standard BASIC EQUATIONS STRATEGY Chapter Simultaneous Eqaations, Solving by Combination Alternatively, you can solve simultaneous equations by combination subtract the two equations to eliminate one of the variables Solve the following for In this method, add or x and y x+y=9 2x= Sy+ Line up the terms of the equations Use combination whenever it is easy to manipulate the cqlWions so that the codBcients fur one variable are the SAME or x+y=9 2x- 5y= If you plan to add the equations, multiply one or both of the equations so that the coefficient of a variable in one equation is the OPPOSITE of that variable's coefficient in the other equation If you plan to subtract them, multiply one or both of the equations so that the coefficient of a variable in one equation is the SAME as that variable'scoeffldenr in the other equation -2 (x + y= 9) 2x- 5y= -2x-2y=-18 2x- 5y= OPPOSITE Note that the x coefficients are now opposites Add the equations to eliminate one of the variables -2x-2y =-18 + 2x- 5y = -7y=-14 Solve the resulting equation for the unknown variable -7y=-14 y=2 Substitute into one of the original equations to solve for the second variable x+y=9 x+2=9 x=7 9danliattan·6MAVPrep the new standard 15 Chapter BASIC EQUATIONS STRATEGY Simultaneous Equations: Three Equations The procedure for solving a system of three equations with three variables is exactly the same as for a system with two equations and two variables You can use substitution or combination This example uses both: Solve the following for WI XI and y Solve three simultaneous equations step-by-step Keep careful track of X+W=y 2y+ W= 3x- 13 - 2w=x+ Y your work to avoid careless errors, and look for The first equation is already solved for y ways to reduce the number of steps needed to y=x+w solve Substitute for y in the second and third equations Substitute for y in the second equation: 2(x+ w) + w= 3x- 2x + 2w + w = 3x - -x+3w=-2 Substitute for y in the third equation: 13 - 2w = x + (x + w) 13 -2w=2x+ 3w+ 2x= 13 w Multiply the first of the resulting two-variable equations by (-1) and combine them with addition x- 3w= + 2x+3w=13 3x= 15 Therefore, x = Use your solution for x to determine solutions for the other two variables 3w+ 2x= 13 3w+ 10 = 13 3w=3 w=1 y=x+w y=5+1 y=6 The preceding example requires a lot of steps to solve Therefore, it is unlikely that the GMAT will ask you to solve such a complex system-it would be difficult to complete in two minutes Here is the key to handling systems of three or more equations on the GMAT: look for ways to simplify the work Look especially for shortcuts or symmetries in the form of the equations to reduce the number of steps needed to solve the system :M.anliattanG MAT'Prep 16 the new standard INACTION AoomONAL VIes PROBLEMSET Chapter 13 Problem Set (Advanced) If b 30 -6 = (A) 12b , then what is a? (S) 4b - xy+xz (S) 10,000 (S) l00(x:y) (0) 100(-y-) ,x-y (E) l00x-y y x, y, and z? xy+xz xyz 100 (E) (e) 100(X~Y) (S) l00yz x (e) l00y xz (0) 10,ooox yz (E) 10,oooyz x If a = 20bc, then a is what percent of b? (S) 2,OOOc (e) c E (0) 2,000 20 (E) c+ 20 If a, b, and c are greater than and a is twice as large as b percent of terms of band c, what is a percent of c? (A) 2bc 100 2bc (S) 1,000 be' (e) 5,000 (0) b2 _c_ 5,000 (E)' c, then in ooob c2 X percent of Y percent of Z is decreased by Y percent What is the result? XZ- 100XYZ - XY2Z (A) - (S) 1,000,000 (0) +6 x is what percent of y percent of z, in terms of x, y and z? (A) 20c of (0) 100 10,000 (A) 100( X;y) z z, in terms xyz+l00xz (e) (E) 3b x is what percent greater than y, in terms of x and y? (A) 100xy (0) 4b +6 What number is x percent of y percent of xyz (A) 10,000 +6 (e) 3b 100 Y XZ-Y (e) 10,000 XYZ- 2Y XYZ- 2Y (E) 100 10,000 Two wooden boards have the same area One of the boards is square and the other is rectangular If the square board has a perimeter of p meters and the rectangular board has a width of w meters, what is the length of the rectangular board, in terms of p and w? (A) E w (S) L 4w (e) L 16w (0) p w (E) p w 16 :M.anFiattanG MAT"Prep the new standard 195 Chapter 13 Bradley owns b video game cartridges If Bradley's total is one-third the total owned by Andrew and four times the total owned by Charlie, how many video game cartirdges the three of them own altogether, in terms of b? 16 (A) -b 10 11 17 (B) -b 13 (C) -b 19 (0) -b 12 (E) -b 12 Linda and Angela contract to paint a neighbor's house Even though Linda spends 50% more time painting the house than Angela, each receives a payment of m dollars when the work is completed If Angela decides to pay Linda dollars so that they would have received the same compensation per hour worked, what is n in terms of m? (A) -m (B) -m (C) -m (0) -m n (E) -m A park ranger travels from his base to a camp site via truck at r miles per hour Upon arriving, he collects a snowmobile and uses it to return to his base If the camp site is d miles from the park ranger's base and the entire trip took t hours to complete, what was his speed on the snowmobile, in terms of t, d and r? (A) tr-d (B) td-r 9rlanliattanGMAT*Prep 196 INACTION ADDITIONAL VIeS PROBLEMSET the new standard dr rt-d (C) drt dt-r (0) td-r d (E) IN ACTION ANSWER KEY AoomONAL VIes SOLUTIONS Chapter 13 (D): Using the Direct Algebra strategy, we solve for a and see which answer choice matches our result: 4b+6 =3a 3a-6 b= 4b+6 -=a 4b = 3a-6 (A): Using the principles laid out for Percent VICs in this chapter, this problem can be solved relatively easily using the Direct Algebra approach: "x percent of " translates to "~x 100 trans Iates to "y- C" x z "Th ererore x percent 100· f y percent .", and "y percent of z" \ f z "Itrans ates to - x x - Y x z = xyz 100 100 10,000 (C): This problem can be solved by using the Direct Algebra approach "x is what percent greater than y' can be solved by creating an intermediate variable, such as to, to represent the percent of y that x is greater than, and then solve for w Recall that "percent greater than y" means plus a percent of y": "» w x=y+-xy 100 x- y=C~O)Y Alternatively, you could use the Hybrid Method Say x = 50 and y = 40 Then x is 25 percent greater than y How we use 50 and 40 to arrive at 25? Simply subtract 40 from 50, divide (50 - 40) = 10 by 40, 50-40) 40 =100 (x-,- then multiply by 100: 100 ( y) (D): This problem can be solved by using the Direct Algebra approach "x is what percent of y percent of z" can be broken down into two components First, "y percent of z" translates to " Lx 100 z " Second, "x is what percent of y percent of z" can be solved by creating an intermediate variable, such as w, to represent the percent of y percent of z" that x equals, and then solve for that variable: x= w(Ifo) 100x yz 10,000x = -=w =w 100 yz 100 (B): Let us use the Pick Numbers and Calculate a Target approach Select values for a, b, and e that make the equation a = 20be true, such as a = 120, b = 2, and e = Since 120 + 60, a is 6,000% of b Test each answer choice to find the one that yields the Target value of 6,000 = (A) 20e = 20 x = 60 Incorrect (B) 2,000e = 2,000 x = 6,000 CORRECT 9danliattanG MAT·prep the new standard 197 Chapter 13 IN ACTION ANSWER KEY ADDITIONAL VIeS SOLUTIONS (C) ;0 = ;0 Incorrect (D)_c 2,000 2,000 Incorrect (E) c + 20 = Incorrect + 20 = 23 (C): Let us use the Pick Numbers and Calculate a Target approach Select values for a, b, and c such that a is twice the size of b percent of c For example, a = 80, b = 25, and c = 160 Since 80% of 160 = 128, we should test each answer choice to find the one that yields the Target value of 128 (A) 2bc 100 (B) (C) = 2bc2 2(25)(160) 100 1,000 E: = 5,000 5,000b (E) -c-2 - = = 25(160)2 5,000 = (160)2 200 (25)2160 5,000 = 25(160) 200 5,000(25) (16W = 80 50(160) 100 1602 - -2 2(25)(160)2 1,000 (D) ~ 5,000 = = Incorrect = 12,800 = 128 CORRECT = 20 50(25) 162 = Incorrect 625 128 Incorrect Incorrect (A): Let us use the Pick Numbers and Calculate a Target approach First, assign numbers to represent X Y, and Z We should pick numbers that translate easily into percents, such as those in the table to the right, because solving this problem will require heavy computation variable Y% of Z = 50% of200 = 100 X% of Y% of Z = 10% of 100 = 10 Decreasing this result by Y%, or by 50%, yields Next, test each answer choice to find the one that yields the Target value of 5: (A) 100XlZ - XY2Z 1,000,000 = 100(10)(50)(200) - (10)(2500)(200) 1,000,000 (B) XZ - Y = (10)(200) - 50 100 100 19 CORRECT Incorrect (C) XZ - Y = (10)(200) - 50 = 0.19 10,000 10,000 Incorrect (D) XlZ - 2Y = (10)(50)(200) - 2(50) = 999 100 100 Incorrect (E) XlZ - 2Y 10,000 = (10)(50)(200) -2(50) 10,000 = 9.99 9danliattanG MAT·Prep 198 =5 the new standard Incorrect number X 10 Y Z 50 200 IN ACTION ANSWER KEY AoomONAL Chapter 13 VIes SOLUTIONS Note that if you find the calculations taking too long to complete, you could stop and choose (A), because it matched our target This ignores the possibility that (A) matched the target by coincidence and some other answer choice is correct However, it may be better to take a good chance on (A) and conserve time than to spent a large amount of time to prove that (A) is correct This strategy would not be as effective if the correct answer were not one of the first answer choices evaluated (C): Let us solve this problem using a Pick Numbers and Calculate a Target approach First, draw a diagram to represent the problem Then assign numbers to represent p and w We should try to pick numbers that will result in the length being an integer, if possible variable (A) i = 24 w (B) = 288 w 242 = -=72 4w 4(2) 6 Incorrect i 242 = =18 16w 16(2) CORRECT (D) iw = 242(2) = 288 Incorrect (E) iw = 242(2) = 72 16 Incorrect 16 Incorrect L (C) - 24 p If the perimeter of the square is 24, then each side of the square must equal Therefore the area of the square will be 62 = 36 If the width of the rectangle is 2, then the length of the rectangle (shown as L) must be 18 so that the rectangle also has an area of 36 Test each answer choice to find the one that yields the Target value of 18: number L 2 L (B): Let us solve this problem using a Pick Numbers and Calculate a Target approach Imagine that Bradley owns 12 video game cartridges Then Andrew owns 36 video game cartridges and Charlie owns video game cartridges In total, the three boys own 12 + 36 + = 51 video game cartridges Therefore we should test each answer choice to find the one that yields the Target value of 51: 16 (A) - 16 b = - (12) 3 = 64 (B) 17 b = 17 (12) = 51 4 Incorrect CORRECT (C) !ib = !i(12) = 39 Incorrect (0) !2.b = = 19 Incorrect =7 Incorrect 12 (E) ~b 12 19 (12) 12 = ~(12) 12 You should take a glance at the relationships before picking numbers For example, since Bradley owns times as many video game cartridges as Charlie, Bradley's total should bea number divisible by (such as 12) 9rf.anhattanG M.AT·Prep the new standard 199 Chapter 13 AoomONAL IN ACTION ANSWER KEY VIes SOLUTIONS 10 (D): For this problem, we can use the Hybrid Method The problem contains two variables and an implicit equation, so we have to pick a value for one of the variables and solve for the other variable We will also invent a variable to represent the number of hours Linda and Angela worked Let us assume m, the original payment in dollars to both Linda and Angela, equals $50 If Angela worked x hours, then Linda worked 1.5x hours Therefore, pick x = (to produce integers for both Angela's hours and Linda's hours) Angela has to make a payment to Linda of n dollars so that both Linda and Angela earn the same hourly wage · d'as hourlv by: -350 + n = -3m +n L10 our y waze wage iIS given 50-n Angela's hourly wage is given by: m+n = Therefore, we have: m+n = m-n 5n=m 2m + 2n = 3m - 3n n=-m 11 (C): This is a Rates & Work problem that can be solved using an RTD Chart We will use the Direct Algebra method to think through the problem Additionally, we will invent a variable, q, to represent the park ranger's speed on the snowmobile: Rate x (units) (miles/hour) Truck r x Snowmobile q x x = Distance Distance = (hours) Total Since Rate x Time = Distance, Time Time (miles) d r d q t = d = d = 2d -;- Rate Therefore we know that the time that the park ranger spent in the truck was d hours and the time he spent driving the snowmobile was d hours q r d d Furthermore, d those times have to sum to the total time for the trip, so r d -+-=t r q dq+dr = rqt q(d +rt) = -dr + - = t We can now solve for q: q -dr q= =-d -rt dr rt-d For more on solving Rates & Work problems, see the "Rates & Work" chapter of the Manhattan W0rd TranslationsStrategy Guide 9danliattan G M AT·Prep 200 the new standard GMAT Chapter 14 -=-0/- +' EQUATIONS, INEQUALITIes, & VIes OFFICIAL GUIDE PROBLEM SETS: PART II In This Chapter • Equations, Inequalities, & VICs Problem Solving List from Th.e Ofjicial Guides: PART II • Equations, Inequalities, & VICs Data Sufficiency List from The Ofjicial Guzdes: PART II OFFICIAL GUIDE PROBLEM SETS: PART II Chapter 14 Practicing with REAL GMAT Problems Now that you have completed Part II of EQUATIONS, INEQUALITIES & VICs it is time to test your skills on problems that have actually appeared on real GMAT exams over the past several years The problem sets that follow are composed of questions from three books published by the Graduate Management Admission Council- (the organization that develops the official GMAT exam): The Official Guide for GMAT Review, 12thEdition The Official Guide for GMAT Quantitative Review The Official Guide for GMAT Quantitative Review, 2nd Edition Note: The two editions of the Quant Review book largely overlap Use one OR the other These books contain quantitative questions that have appeared on past official GMAT exams (The questions contained therein are the property of The Graduate Management Admission Council, which is not affiliated in any way with Manhattan GMAT.) Although the questions in the Official Guides have been "retired" (they will not appear on future official GMAT exams), they are great practice questions In order to help you practice effectively, we have categorized every problem in The Official Guides by topic and subtopic On the following pages, you will find two categorized lists: (1) Problem Solving: Lists MORE DIFFICULT Problem Solving Equations, Inequalities & VIC questions contained in The Official Guides and categorizes them by subtopic (2) Data Sufficiency: Lists MORE DIFFICULT Data Sufficiency Equations, Inequalities & VIC questions contained in The Official Guides and categorizes them by subtopic Each book in Manhattan GMAT's 8-book strategy series contains its own Official Guide lists that pertain to the specific topic of that particular book If you complete all the practice problems contained on the Official Guide lists in each of the Manhattan GMAT strategy books, you will have completed every single question published in The Official Guides :ManliattanG M.AT*Prep the new standard 20S Chapter 14 OFFICIAL GUIDE PROBLEM SOLVING SET: PART II Problem Solving: Part II from The Official Guidefor GMAT Review, ir: Edition (pages 20-23 & 152-185), The Official Guide for GMAT Quantitative Review (pages 62-85), and The Official Guide for GMAT Quantitative Review, 2nd Edition (pages 62-86) Note: The two editions of the Quant Review book largely overlap Use one OR the other Solve each of the following problems in a notebook, making sure to demonstrate how you arrived at each answer by showing all of your work and computations If you get stuck on a problem, look back at the EQUATIONS, INEQUALITIES, and VIes strategies and content contained in this guide Note: Problem numbers preceded by "D" refer to questions in the Diagnostic Test chapter of The Official Guide for GMAT Review, 1Zh edition (pages 20-23) ADVANCED SET - EQYATIONS, INEQUALITIES, & VICs This set picks up from where the General Set in Part I left off Basic Equations 12th Edition: 196,218 Quantitative Review: 155, 173 Equations with Exponents 12th Edition: 150, 172 Quantitative Review: 153, 166 OR 2nd Edition: 166 Quadratic Equations 12th Edition: 215, 222 Quantitative Review: 121 OR 2nd Edition: 121 Formulas & Functions 12th Edition: 146, 171,228 QR 2nd Edition: 67, 131, 158 Inequalities 12th Edition: 161, 173 QR 2nd Edition: 156 VICs 12th Edition: 163, 165,202,204,208,212,213,227 Quantitative Review: 118, 124, 128, 133, 146, 171, 172 OR2ndEdition: 118, 124, 128, 133, 171, 172 CHAllENGE SHORT SET - EQUATIONS, INEQUALITIES, & VICs This set covers Equations, Inequalities, and VIC problems from each of the content areas, including both easier and harder problems, but with a focus on harder problems The Challenge Shorr Set duplicates problems from the General Set (in Pan I) and the Advanced Set above 12th Edition: 34, 38,41,58,68,89, 122, 130, 144, 146, 163, 173, 188, 196,204,208,212, 213,215,222,228, D16, D24 Quantitative Review: 52, 83, 85, 92, 104, 106, 107, 153, 155, 173 OR 2nd Edition: 41,54,85,92, 103, 104, 106, 107, 131, 156 9rtanfiattanG 204 MAT·Prep the new standard OFFICIAL GUIDE DATA SUFFICIENCY SET: PART II Chapter 14 Data Sufficiency: Part II from The Official Guide for GMAT Review, 12th Edition (pages 24-26 & 272-288), The Official Guide for GMAT Quantitative Review (pages 149-157), and The Official Guide for GMAT Quantitative Review, 2nd Edition (pages 152-163) Note: The two editions of the Quant Review book largely overlap Use one OR the other Solve each of the following problems in a notebook, making sure to demonstrate how you arrived at each answer by showing all of your work and computations If you get stuck on a problem, look back at the EQUATIONS, INEQUALITIES, AND VIes strategies and content contained in this guide Practice REPHRASING both the questions and the statements by manipulating equations and inequalities The majority of data sufficiency problems can be rephrased; however, if you have difficulty rephrasing a problem, try testing numbers to solve it It is especially important that you familiarize yourself with the directions for data sufficiency problems, and that you memorize the fixed answer choices that accompany all data sufficiency problems Note: Problem numbers preceded by "D" refer to questions in the Diagnostic Jest chapter of The Official Guide for GMAT Review, 12!h edition (pages 24-26) ADVANCED SET - EQUATIONS, INEQUALITIES, & VIes This set picks up from where the General Set in Pan I lefr off Basic Equations 12th Edition: 95, 125, 150, 168, D37 Quantitative Review: 92, 102, 103, 118 OR 2nd Edition: 106, 124 Equations with Exponents Edition: 165 Quantitative Review: 105, 115 OR 2nd Edition: 109, 121 i» Quadratic Equations Edition: 158 Quantitative Review: 79, 80 OR 2nd Edition: 83 i» Formulas & Functions Edition: 115 QR 2nd Edition: 107, 111 i» Inequalities 12th Edition: 97, 153, 162, D38 Quantitative Review: 66, 67, 85, 114 OR 2nd Edition: 68, 69, 89, 120 CHALLENGE SHORT SET - EQUATIONS, INEQUALITIES, & VICs This set covers Equations, Inequalities, and VIC problems from each of the content areas, including both easier and harder problems, but with a focus on harder problems The Challenge Short Set duplicates problems from the General Set (in Part I) and the Advanced Set above 12th Edition: 26, 30, 38, 45,80,97, 115, 125, 154, 156, 158, 162, 168, D30, D33 Quantitative Review: 85, 115 OR 2nd Edition: 89, 121 :M.anliattanG MAT·Prep the new standard 205 Chapter By Chapter 9vLanliattan GMAT PART I: GENERAL BASIC Part of 8-Book Series Number Properties Fractions, Decimals, & Percents Equations, Inequalities, &VICs Word Translations Geometry Critical Reasoning Reading Comprehension Sentence Correction EQUATIONS: One-Variable Equations, Simultaneous Equations, Mismatch Problems, Combo Manipulations, Absolute Value Equations EQUATIONS WITH EXPONENTS: Even Exponents, Common Bases,Common Exponents, Eliminating Roots 1QUADRATIC EQUATIONS: Factoring, Disguised Quadratics, Foiling, 1-Solution Quadratics, Undefined Denominators, Special Products FORMULAS: Plug-Ins, Strange Symbols, Unspecified Amounts, Sequences, Patterns FUNCTIONS: Numerical Substitution, Variable Substitution, Compound Functions, Unknown Constants, Graphs, Common Function Types INEQUALITIES: Combining Inequalities, Compound Inequalities, Extreme Values, Optimization, Testing Cases,Inequalities and Absolute Value, Square-Rooting 7.VICs: Variables In Answer Choices, Examples, Strategies, Pros and Cons of Different Strategies PART II: ADVANCED Includes separate chapters on Advanced Equations, Inequalities, & VICs topics, as well as additional practice problem! What's Inside This Guide • Clear explanations of fundamental principles • Step-by-step instructions for important techniques • Advanced chapters covering the most difficult topics • In-Action practice problems to help you master the concepts and methods • Topical sets of Official Guide problems listed by number (problems published separately by GMAC) to help you apply your knowledge to actual GMAT questions • One full year of access to Computer Adaptive Practice Exams and Bonus Question Bank How Our GMAT Prep Books Are Different • Challenges you to more, not less • Focuses on developing mastery • Covers the subject thoroughly Comments • Not just pages of guessing tricks • Real content, real structure, real teaching • More pages per topic than all-in-l tomes From GMAT Test Takers "I've loved the materials in the Strategy Guides I've found I really learned a lot through them It turns out that this was the kind of in-depth study and understanding that I needed The guides have sharpened my skills I like how each section starts with the basics and advances all the way through the most complicated questions." "The material is reviewed in a very complete and user-friendly in a way that gets to the heart of the matter by demonstrating a very thorough and uncumbersome fashion." ", , :';.' I=In4U5 onUne at: www.manhattangmat.com manner The subjects are taught how to solve actual problems in ,& Canada: 1.800.5'16.4628 rnational: 001.212.721.7480 ... left-hand side of the equation is the first key to isolating the combo on one side of the equation; then we have to subtract 2ac from both sides of the equation 9danliattanG MAT 'Prep the new standard... as usual there are two solutions for x and y (x = and y = 8, or x -5 and y -8) , so the statements are NOT SUFFICIENT, even together (answer E) = = Combo Problems: Manipulations The GMAT often asks... should try to manipulate the given equation(s) in either the question or the statement, so that the combo is isolated on one side of the equation Then, if the other side of an equation from a statement