Considered as one of the most coveted entrance exams, the Graduate Management Admission Test (GMAT) examines the basic analytical, quantitative and verbal skills of the candidates. Although every section has its own significance, the quantitative section of GMAT is the trickiest one. Quantitative Aptitude examines your basic knowledge of mathematics concepts. One such key topic of the Quants section is Inequalities Questions. Since the questions asked from this topic carry significant weightage in the exam, it becomes important to understand this concept thoroughly. Here is a comprehensive blog detailing Inequality questions, how you can solve them by following some basic rules as well as various sample questions to help you grasp the basics of this concept.
This Blog Includes:
- What are Inequalities?
- Types of Inequality Questions
- Rules For Solving Inequality Questions
- How to Solve Inequalities Questions for GMAT?
- How to Solve Absolute Value Inequalities in GMAT?
- Important Points to Remember
- Quick Tips for Inequality Questions for GMAT
- Wavy Curve Method to Solve Inequality Questions for GMAT
- Sample Inequality Questions for GMAT
What are Inequalities?
Before moving on to the fundamental formulas for inequality questions, it is pivotal to simplify the concept of inequalities. They are defined as the functions which determine a relationship between distinct values and expressions. This crucial feature differentiates inequalities from equations. While equations represent the relationship between equal values, inequalities compare unequal quantities. The symbols which represent inequalities are : ‘>’, ‘<‘, ‘≥’, ‘≤’ and ‘≠’.
X ≠ Y
Here, ‘≠’ is called as the ‘unequal’ sign. We may not know which variable out of X or Y has a higher or a lower value but they are certainly not equal.
X > Y
The inequality is referred to as the ‘greater than’ sign. As per the above equation, X is greater than Y.
X < Y
This symbol is the opposite of greater than and is called as ‘less than’. Thus, X is less than Y.
X ≥ Y
The combination of greater than and equal to sign shows that the variable X is greater than Y but can also be equal.
X ≤ Y
This is the opposite of greater than equal to sign. Thus, variable Y can be greater than or equal to X.
Types of Inequality Questions
Generally, the inequality questions for GMAT are of two types, i.e.
1. Direct Inequalities – The questions on direct inequalities are comprised of the relationship between variables which is indicated in the symbol form. Example: P>Q=R<S
2. Indirect Inequalities – When it comes to question on indirect inequalities, the relationship between variables is defined in a coded form. Different symbols like @, #, %, $ etc. are used to represent this relationship.
Rules For Solving Inequality Questions
Inequalities can seem tricky and complex if you don’t know about the basic rules of solving them. Here is a list of some key rules and tips you can follow to simplify inequality questions:
1. When you divide or multiply an inequality by a negative number, the inequality sign flips.
Example: 12 > – 5
On multiplying both sides with -5, we get the equation as -60 > 25. But, – 60 cannot be greater than 25. Thus, it becomes important to reverse the symbol.
2. In Inequality Questions where symbols are opposite, no conclusion can be drawn.
Example: in P>Q<R, it cannot be concluded whether P is greater than R or vice versa.
3. In complementary pairs, the relationship cannot be established if the relation between common elements is not defined.
Example: In P ≥ Q, Q ≤ R, Q is less than or equal to both P and R. Thus, the relationship between P and R cannot be established.
4. The inequality direction does not change if we add or subtract numbers on both sides.
Example: On adding 2 to both sides of 10 < 15 we get 12 < 17. Which stands true in both the cases.
5. Dividing the inequality with a variable whose sign is unknown does not yield the right answer.
Example: PX < 3P. On dividing the inequality with P on both sides we get, X < 3.
But if we divide it with -P then we get -X < -3.
Thus, it is advisable to not divide the inequality with a variable whose sign remains unknown.
How to Solve Inequalities Questions for GMAT?
We have inequalities easy for you. Learn the how-to solve inequalities questions step by step for acing your GMAT exam:
- Terminate the fractions in inequalities by multiplying all terms by the least common denominator of all fractions.
- To Simplify the inequalities, combine like terms on each side of the inequality.
- In order to obtain the unknown on one side and the numbers on the other, start adding or subtracting the quantities
- The last step is to divide each term of the inequality by the coefficient of the unknown.
- Remember inequality remains unchanged if the coefficient is positive and inequality will be reversed if the coefficient is negative.
How to Solve Absolute Value Inequalities in GMAT?
Most of the students find it difficult to solve the absolute value inequalities questions in GMAT and often skip it. You don’t have to worry about this, Most of the absolute value inequalities will have two solutions and here are super easy steps to solves the absolute value inequalities.
- Separate the absolute value expression in the inequality.
- Set an expression equal to both positive and negative values.
- Now, follow the same steps you would for inequalities.
Important Points to Remember
Now that you are familiar with the types and rules of inequality questions for GMAT, Here are some important points that will help you while solving the equations:
- Don’t change the inequality sign while adding or subtracting any quantity on both sides of the inequality.
- Don’t change the changing inequality sign while multiply or dividing by a positive value.
- Always flip the inequality sign if you are multiplying or dividing by a negative number.
- If the quantities are both positive then remember to square both sides.
- In case the sign of the variable is unknown, multiply or divide both quantities by a variable.
Quick Tips for Inequality Questions for GMAT
Inequality questions for GMAT are not easy to solve but here are some quick tips that will help you in the GMAT exam:
- If you feel the inequality questions are hard to solve or tricky, remember the algebra you learned back in school. Same algebra rules are applied for inequality questions as well.
- One simple mistake every student possibly makes is not flipping the sign while multiplying or dividing by a negative number. Always remember to flip the sign if you are multiplying or dividing on each side.
- Never subtract, multiply or divide two sets of inequalities. Two sets of inequalities can only be added.
Wavy Curve Method to Solve Inequality Questions for GMAT
The wavy Curve method is used for solving most of the inequality questions for GMAT. Students initially feel confused. Here is a stepwise easy method to learn the wavy curve method:
- To identify the range of the variable in the given question, draw a horizontal number line.
- Mark the zero points and start to draw a wavy line from the top right portion.
- The wavy line will pass through the corresponding zero points if the terms are odds and in the case of even terms, the wavy line bounces off the corresponding zero points.
- To identify the correct range of values, always remember the value of the expression is positive for regions above the number line, the Value of expression equal to zero on intersection points, and the value of the expression is negative for regions below the number line.
Sample Inequality Questions for GMAT
By now, you must have understood the basics of Inequalities and its different concepts. Here are some sample Inequality Questions which you can practice to strengthen the basics:
1. If 3 < x < 8 and 5 < y < 11, which of the following represents all of the possible values of xy?
(A) 3 < xy < 11
(B) 8 < xy < 19
(C) 15 < xy , 88
(D) 24 < xy < 55
2. Consider three distinct positive integers a, b, c all less than 100. If |a – b| + |b – c| = |c – a|, what is the maximum value possible for b?
3. Consider integers p, q, r such that |p| < |q| < |r| < 40. P + q + r = 20. What is the maximum possible value of pqr?
4. If −1 < x < 5, which is the following could be true?
(A) 2x > 10
(B) x > 17/3
(C) x^2 > 27
(D) 2x – x^2 < 0
5. If |(x – 3)2 + 2| < |x – 7| , which of the following expresses the allowable range for x?
(A) 1 < x < 4
(B) 1 < x < 7
(C) – 1 < x < 4 and 7 < x
(D) x < – 1 and 4 < x < 7
(E) – 7 < x < 4 and 7 < x
Flip the inequality while you multiply or divide both sides of the inequality.
When both sides are non-negative you can square both sides of an inequality
Yes, multiplying both sides by the denominators will cross multiply the inequality.
The wavy Curve method is used for solving inequalities questions.
Absolute inequality will have only two solutions- real numbers or no solutions.
We hope that this blog helped you gain better clarity about the Inequality questions for GMAT. If you are appearing for GMAT and need more tips and tricks for solving other sections, our Leverage Edu experts can guide you throughout the preparation process and helping you ace every section on this exam and successfully clearing it with high scores.