Inequalities(Algebraic equations) Shortcut method with solved examples

Inequalities

These types of questions generally come in bank exams. In these questions, you have to calculate and compare the roots of the quadratic equations.

In this post, I am using the shortcut method only to calculate and compare the roots.

We have to compare two roots x and y.

Comparison:

1. x > y, x is greater than y.

2. x < y, x is less than y

3. x ≥ y, x is greater than or equal to y

4. x ≤ y, x is less than or equal to y

5. x = y, x is equal to y

6. CND, Cannot determine if we have two roots x1 > y1 and x2 < y2 simultaneously.

 

Most important formulas for inequality:

The equations will be given in the form of: 

x²±ax±b = 0

y²±cy±d = 0

Then,

The following table shows the signs of the Roots of equations:

Question	Answer
+ +	- -
- +	+ +
+ -	- +
- -	+ -

Solved examples:

Ex1. x² +7x +12 = 0

        y² -5y + 6 = 0

Sol.Signs of Cofficients in question is 

+ +

- - 

Thus roots will have signs

-    -

+   +

Now, 

For equation i. Factors of 12 whose addition are 7 are 4 and 3.

Roots x become, -4 and  -3 Imp. The larger root will be written first.

For equation ii.  factors of 6 whose addition are 5 are 3 and 2

Roots y  are  +3,+2.

now -4 < +3, -4 < +2

-3 < +3 , -3 < +2

Thus x < y is the answer.

Imp. In this question, you can answer by looking at the signs of the roots no need to calculate values.

negative roots will always less than positive roots thus x < y

From onwards I am saving questions in few lines

Ex.2 x²-6x-16

        y²+6y-16

 Sol.Coffecients have signs,

 - -

+ -

Roots will have signs, 

+  -

-   + 

Factors of the equations i and ii are  8,2 and 8,2

x = +8, -2

y = -8, +2

 x>y,     x>y   and x>y,    x<y  this means Solution is cannot determine CND.

+8>-8, -2>-8 and -2>-8, -2<+2

 

Ex.3 x² +21x-46

        y² + 2y -81

Sol. Signs of coefficients are 

  + - 

  + -

Roots will have signs 

- + 

- +, Means solution is CND

No need to calculate roots.

Ex.4 21x² - 17x +2  ....i

        56y² - 15y +1   ....ii

Sol. Roots will have signs ++ and ++

For equation i,

21*2 = 42, Factors that add up to 17 and multiplied to 42 are 14 and 3

The roots of the equation are +14/21 and +3/21

For equation ii,

56*1 = 56, Roots that add up to 12 and multiplied to 56 are 8 and 7

The roots of the equation are +8/56 and +7/56.

It is difficult to compare these roots, therefore there is a short method to save time

7 is common in 21 and 56,

Divide them by 7, we will get

21/7 = 3 and  56/7 = 8

Now multiply 3 by the solution of equation ii, and 8 by the solutions of the equation i.

we get roots of equations.

+14*8,+3*8 and 8*3, 7*3

+112,+24 and 24,21

The solution is x>y

Type 2. When we have to calculate the value of x and y of order n, like xⁿ  and yⁿ

If order is even then value will be  + and -

If order is odd then value will be +

Ex1. x² = 16

        y² = 16

Sol. x = +4,-4

       y = +4,-4

Solution is cannot determine.

Ex.2 x⁴ = 81

        y³ = 64

Sol x = +9,-9

      y= +8

x < y Ans

Ex.3 x² = 16

        y = √16

Sol. x = +4,-4

       y = +4