Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized for You
we will pick new questions that match your level based on your Timer History
Track Your Progress
every week, we’ll send you an estimated GMAT score based on your performance
Practice Pays
we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:
Join Malvika Kumar, Asst. Director (Admissions) IESE, to learn about the career prospects and more. Understand how an MBA from IESE can help transform your career by developing essential managerial skills such as leadership, data-driven decision making.
Whether you’re putting the finishing touches on your Round 1 application or you’re just getting started on your application for Round 2, this online chat will provide all of the insider tips and advice you need to help you prepare!
Working in collaboration with examPAL we will provide you with a unique online learning experience which will help you reach that higher score. Start your free 7 day trial today.
This session addresses the queries often raised by too young or too old MBA candidates about their acceptability in MBA programs, post MBA careers, and substitutes for MBA.
Does GMAT RC seem like an uphill battle? e-GMAT is conducting a free webinar to help you learn reading strategies that can enable you to solve 700+ level RC questions with at least 90% accuracy in less than 10 days.
GMAT Club Presents MBA Spotlight October 2020 - a dedicated Business School MBA Fair that brings MBA applicants and World’s Top Business Schools together for a fun and unique experience of live presentations, Q&A, and prizes!
Attached pdf of this Article as SPOILER at the top! Happy learning!
Hi All,
I have recently uploaded a video on YouTube to discuss Inequalities (Basic) in Detail:
Following is covered in the video:
Theory
What is Inequality and Types of Inequalities Graphing Inequalities Properties of Inequalities Types of Inequality Problems x*y > 0 x/y > 0 x*y < 0 x/y < 0 Basic Problems on Inequalities
After this Post Check out the post on How to Solve Inequality Problems using Two Methods : Algebra and Sine Wave Method / Wave Method or Wavy Method
What is Inequality and Types of Inequalities
Usually we are given discrete values of variables like x=2, y=3 etc. In case of inequalities we are given a range of values. Let's take some examples to understand this:
x > 3 => x can take all real values which are greater than 3, i.e. 3.001, 4, 5, 6, 8, 100, etc... So, instead of giving a single value in case of inequalities we are given a set of values for the variables.
Let's understand various types of inequalities now:
Greater Than Inequality ( > ) : Ex: x > 3 (We have seen above) Less Than Inequality ( < ) : Ex: y < 2 => Y can take all read values which are less than 2 Greater Than or Equal to Inequality ( ≥ ): Ex: x ≥ 5 (x can take all real values greater than or equal to 5) Less Than or Equal to Inequality ( ≤ ): Ex: y ≤ 3 (y can take all real values less than or equal to 3) In-Between Inequality ( -2 ≤ x < 5 ): Ex: x can take all values which are greater than or equal to -2 and less than 5
Graphing Inequalities
Now let's talk about how to plot and inequality on the number line
Ex 1: Graph a ≤ 2 and b > -3 on a number line Sol: To plot a ≤ 2 we need to draw a line starting at 2 and extending till -∞ on the left hand side (Refer Orange line in below image). Note that we need to darken point 2 because it is included (as a ≤ 2, so 2 is included) To plot b > -3 we need to draw a line starting at -3 and extending till +∞ on the right hand side. (Refer Green line in below image). Note that we DO NOT darken point -3 as it is excluded (As b > -3 and not ≥ -3 )
Properties of Inequalities
PROP 1: Adding or Subtracting the same number from both the sides of the inequality DOES NOT change the sign of the inequality.
Ex 1: 7 > 3 Add 4 on both the sides we get 7 + 4 > 3 + 4 => 11 > 7 [Which is True and Note that sign of inequality which was > is still > ]
Ex 2: 8 > 4 Subtract 9 from both the sides we get 8 -9 > 4 -9 => -1 > -5 [Which is True and Note that sign of inequality which was < is still < ]
Ex 3: a > b Add k on both the sides we get a + k > b + k [ which is true and sign of inequality did not change ]
PROP 2: Multiplying / Dividing an inequality equation with a positive number DOES NOT change the sign of the inequality
Ex 1: 7 > 3 Multiply both the sides by +2 we get 7 * 2 > 3 * 2 => 14 > 6 [ which is true and sign of inequality did not change ]
Ex 2: 8 > 4 Divide both the sides by +2 we get \(\frac{8}{2}\) > \(\frac{4}{2}\) => 4 > 2 [ which is true and sign of inequality did not change ]
Ex 3: a > b Multiply both the sides with a positive variable k we get ak > bk
PROP 3: Multiplying / Dividing an inequality equation with a negative number REVERSES the sign of the inequality
Ex 1: 7 > 3 Multiply both the sides by -2 we get 7 * -2 < 3 * -2 => -14 < -6 [ note the sign of inequality has changed from > to < ]
Ex 2: 8 > 4 Divide both the sides by -2 we get \(\frac{8}{-2}\) < \(\frac{4}{-2}\) => -4 < -2 [ note the sign of inequality has changed from > to < ]
Ex 3: a > b Multiply both the sides with a negative variable t we get at < bt [ note the sign of inequality has changed from > to < ]
PROP 4: We can add two inequalities which have the same inequality sign
Ex 1: 7 > 3 and 8 > 2, Since the two inequalities have same sign of (>) so we can add both of them to get 7 + 8 > 3 + 2 => 15 > 5
Ex 2: a > b and c > d Since the two inequalities have same sign of (>) so we can add both of them to get a + c > b + d [Note that this is true irrespective of the signs of a, b, c and d]
Ex 3: If two inequalities have different signs then we can multiply one of them to make the signs same and then add them a > b c < d we can multiple c < d with -1 to get -c > -d and now we can add a > b and -c > -d to get a - c > b - d
PROP 5: Taking Square Root on both sides of an inequality DOES NOT Change the sign of the inequality (provided it is possible to take square root on both the sides and get real values).
Ex 1: \(a^2\) > \(b^2\) [given that a and b are positive numbers] Taking square root on both the sides we will get a > b
PROP 6: Square of a number is always non-negative
Ex 1: \(a^2\) ≥ 0 [ this is true for all real values of a ] this will be equal to 0 only when a itself is zero
Types of Inequality Problems
Type 1: x * y > 0 If product of two variables > 0 that means that the two variables have SAME SIGN Either Both are Positive => x > 0 and y > 0 Or Both are Negative => x < 0 and y < 0
Type 2: x / y > 0 If division of two variables > 0 that means that the two variables have SAME SIGN Either Both are Positive => x > 0 and y > 0 Or Both are Negative => x < 0 and y < 0
Type 3: x * y < 0 If product of two variables < 0 that means that the two variables have DIFFERENT SIGN Either x > 0 and y < 0 Or x < 0 and y > 0
Type 4: x / y < 0 If division of two variables < 0 that means that the two variables have DIFFERENT SIGN Either x > 0 and y < 0 Or x < 0 and y > 0
Basic Problems on Inequalities
Q1. Is bd > 0 ? A. ab > 0 B. cd > 0
Sol: Stat A: ab > 0 There are two cases a>0 and b>0 a<0 and b<0 In both the cases we don’t know anything about the sign of d so NOT sufficient
Stat B: cd > 0 There are two cases c>0 and d>0 c<0 and d<0 In both the cases we don’t know anything about the sign of b so NOT Sufficient
Combining both the statements we will have four cases (1) a>0 b>0 c> 0 d>0 (2) a>0 b>0 c< 0 d<0 (3) a<0 b<0 c>0 d>0 (4) a<0 b<0 c<0 d<0 In case 1 and 4 bd > 0 and in case 2 and 3 bd < 0 So, Together also NOT sufficient. So, Answer will be E
Q2. Is bd>0 ? A. ab > 0 B. ad > 0
Sol: Stat A: ab > 0 There are two cases a>0 and b>0 a<0 and b<0 In both the cases we don’t know anything about the sign of d so NOT sufficient
Stat B: ad > 0 There are two cases a>0 and d>0 a<0 and d<0 In both the cases we don’t know anything about the sign of b so NOT Sufficient
Combining both the statements we will have two cases (Since we have a common variable “a” in both the statements so we will combine the two statements based on the sign of the common variable First case of STAT A will be combined with the first case of Stat B and Second case of STAT A will be combined with the second case of Stat B (1) a>0 b>0 d>0 (2) a<0 b<0 d<0 In both the cases bd > 0 So, Together the two statements are sufficient. So, Answer will be C
Q3. Given that y = 4 + \((1-x)2\). Find yMin( Minimum Value of y) and find the value of x for which y = yMin
Sol: y = 4 + \((1-x)^2\) We know that square of a number can never be negative, so Min value of \((1-x)^2\) = 0 when 1-x = 0 or when x = 1 So, yMin = 4 when x = 1
This post was shared to cover the basic of inequalities, check out the next post of How to Solve Inequality Problems to learn two methods of solving inequality Problems _________________
GMAT ONLINE Rules are changing on Sept 23: - 1 Retake is allowed - Price is going up to $250 - Change and reschedule Fees are Back - Tests now count towards lifetime and annual test limits