Inequality quantitative questions are very common in every govt competitive exam. This is an important topic and there is a high chance of coming inequality quantitative questions in the exam like IBPS PO, SBI PO, IBPS RRB, IBPS Clerk, SBI Clerk, IBPS RRB, SSC CGL, SSC CGL, SSC MTS, NICL AO, LIC AAO, SBI Associate Clerk, SBI Associate PO and others.

In this article step-by-step we will discuss important concepts tricks and tips to solve inequality questions in quantitative aptitude. Using these simple techniques and short methods you can solve any types of inequality quantitative questions in minimum time. In our previous article, we have discussed **“How to solve Order & Ranking question in reasoning”**.

For easy understanding of this chapter, We will explain all the concepts and different types of inequality quantitative questions from recent competitive exam.

#### What is inequality in mathematics?

In mathematics, an inequality is a relation that holds between two values when they are different.

Let us consider X and Y as two variable and then inequalities can be shown as follows

- X>Y- X is greater than Y
- X<Y- X is less than Y
- X≥Y- X is greater than or equals to Y
- X≤Y- X is less than or equals to Y

In this type of problem, we have to find inequalities between variables which are given in form of an algebraic equation.

Equations may be linear, quadratic or cubic. For the linear and cubic equations, there will be always a definite inequality between variables. But in the quadratic equation, there are some cases where no inequality exists between variables. In such cases, we mark our answer as “Cannot be determined” or “Relation cannot be established”.

#### Inequalities for Linear Equation

The general form of a linear equation with two variables is aX±bY±c=0

In linear equation, both X and Y have only one value. So relation can be established easily.

X+7Y=9, 2X−Y=18. Multiplying 2^{nd }equation by 7.

X+7Y=9……(i) 14X−7Y=126……(ii)

Subtracting equation(i) from (ii) we get X= 9 and Y= 0. hence, X>Y

#### Inequalities for Squares

The general form of a square equation is X^{2}=C

In this type, variables have both positive and negative value.

X^{2}=225 and Y^{2}=289, so X=±15 and Y=±17

+17 is greater than both +15 and −15, but −17 is less than both +15 and −15. So the answer will be “Cannot be determined”

###### NOTE

If both equations are given in the square form, our answer will be “Cannot be determined”

#### Inequalities for Squares and Square Root

If equations are given in this form X^{2}=C and Y=√C where C is a perfect square, then the value of X=±√C. But in the other equation where Y=√C, we consider only positive value of Y as square root always gives a positive value.

For example,

X^{2}=225 and Y=√289. We know that square root always gives a positive value. So Y will have only +17 not −17. Therefore X=±15 and Y=+17, hence Y>X

#### Inequalities for Cube Equation

The general form of a cubic equation is X^{3}=±C

For example,

If X^{3}=4096 & Y^{3}=1728, then X=16 and Y=12. Hence X>Y.

If X^{3}=−4096 & Y^{3}=1728, then X=−16 and Y=12. Hence Y>X.

###### NOTE

When both equations are cube form. If X^{3}>Y^{3 }then X>Y and Y^{3}>X^{3}^{ }then Y>X.

#### Inequalities for Square and Cube

If X^{2}=25 and Y^{3}=125, then X=±5 and Y=5 by comparing we get Y≥X.

If X^{2}=36 and Y^{3}=64, then X=±6 and Y=4. So Y=4 is greater than X=−6 and less than X=+6. Hence relation cannot be established.

**Solving Technique for Quadratic Equation**

The general form of a quadratic equation is aX^{2}±bX±C=0

In this type of problem two quadratic equations are given from which we have to find the inequalities between roots i.e X and Y. Roots are found by the method of factorization. Step-by-step we will discuss a quick way to find roots from the quadratic equation and inequalities from different types of equations.

**How to find roots from a quadratic equation**

Direct formula to find roots of the quadratic equation, aX^{2}+bX+c=0

But always using this formula is very cumbersome and calculative. Here we have one quicker approach to find roots of any quadratic equation.

Let see an example 3X^{2}+8X+4=0. We have to find the roots of this equation.

- In the first step, multiply coefficient of X
^{2}and the constant term then divide it into two parts in such a way that by adding or subtracting them we get the coefficient of X.For example, the coefficient of X^{2 }is 3 and the constant term is 4. By multiplying them we get 12. By factorizing 12 into 6 and 2 in such a way that by adding them we get 8. - In the next step, divide both factors with the coefficient of X
^{2}and applying appropriate sign from the table we get our roots.

Sign of the roots is dependent on signs before aX and the constant terms(C).

For example, 4Y^{2}−19Y+12=0

Using the method we find Y= 4, ¾

According to the table, both roots are positive as the coefficient of Y is negative and coefficient of constant is positive.

We can determine the inequality by the structure of the equations given in the question.

To simply the concept, equations are divided into two type based on the sign of the product of the coefficient of X^{2 }or Y^{2 }and respective constant terms.

###### Type#1

When both products of the equation are positive and the structure of the equations are like bellow.

1. If equations are like a_{1}X^{2}+b_{1}X+c_{1}=0 and a_{2}Y^{2}−b_{2}Y+c_{2}=0 then the answer is always X<Y

For example, 1. X^{2}+3X+2=0 2. 3Y^{2}−18Y+24=0

Sol. Roots of X= −2, −1 and Y= 2, 4. By comparing roots we get Y>X

2. If equations are like a_{1}X^{2}−b_{1}X+c_{1}=0 and a_{2}Y^{2}+b_{2}Y+c_{2}=0 then the answer is always X>Y

For example, 1. X^{2}−6X+8=0 2. 2Y^{2}+5Y+2=0

Sol. Roots of X= 2, 4 and Y= −2, −½. By comparing roots we get X>Y

###### Type#2

When both products of the equation are negative and if the structure of the equations are like a_{1}X^{2}±b_{1}X-c_{1}=0 a_{2}Y^{2}±b_{2}Y-c_{2}=0, the answer is always “Relation cannot be established”.

- When equations are like a
_{1}X^{2}+b_{1}X−c_{1}=0 and a_{2}Y^{2}−b_{2}Y−c_{2}=0

For example, 1. X^{2}+X−2=0 2. Y^{2}−Y−12=0

Sol. Roots of X= −2, 1 and Y= 4, −3. As X>Y and X<Y, so answer is “cannot be determined”.

2. When equations are like a_{1}X^{2}+b_{1}X−c_{1}=0 and a_{2}Y^{2}+b_{2}Y−c_{2}=0

For example, 1. X^{2}+2X−15=0 2. Y^{2}+2Y−8=0

Sol. Roots of X= −5, 3 and Y= −4, 2. As X>Y and X<Y, so answer is “cannot be determined”.

3. When equations are like a_{1}X^{2}−b_{1}X−c_{1}=0 and a_{2}Y^{2}+b_{2}Y−c_{2}=0

For example, 1. X^{2}−7X−60=0 2. Y^{2}+5Y−66=0

Sol. Roots of X= 12, −5 and Y= −11, 6. As X>Y and X<Y, so answer is “cannot be determined”.

4. When equations are like a_{1}X^{2}−b_{1}X−c_{1}=0 and a_{2}Y^{2}−b_{2}Y−c_{2}=0

For example, 1. X^{2}−3X−40=0 2. 4Y^{2}−16Y−105=0

Sol. Roots of X= 8, −5 and Y= 7·5, −3·5. As X>Y and X<Y, so answer is “cannot be determined”.

###### Type#3

If any other combination of equations is provided except **Type#1** and **Type#2** we have to compare the roots side by side to find the inequality.

In this type, if roots satisfy the following conditions then our answer will be “cannot be determined”.

#1- If the values of X are in between the values of Y or the values of Y are in between the values of Y.

For example,

- 4X
^{2}−24X+27=0 2. Y^{2}−7Y+6=0

Sol. Roots of X= 4·5,1·5 and Y= 6,1

In the above problem values of X are in between values of Y. So the answer is “cannot be determined” as no relation can be established between X and Y.

Let see another example,

- X
^{2}−X−20=0 2. Y^{2}+4Y+3=0

Sol. Roots of X= 5,−4 Y=−3,−1

Similarly, in this example, the values of Y are in between values of X as the previous example so the answer is “cannot be determined”.

#2- If the values of X and values of Y are overlapped which means any value of X or Y are in between value of X and Y.

For example,

- X
^{2}−3X−18=0 2. Y^{2}−13Y+40=0

Sol. Roots of X= 6,−3 Y= −8,−5

In this example, the value of X is in between of Y or the value of Y is in between the value of X. So the answer is “relation cannot be established”.

Finally, you have learned all the concepts and tricks to solve inequality quantitative questions in the upcoming exams.

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