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Inequality / Quadratic Equation
Concept of Inequality and Quadratic Equation
In most of the banking and insurance exam Inequality comes. It is an important topic for a preliminary exam. Most of the time 5 marks come from this topic. You can easily score these 5 marks if you can study this chapter and practice this type of question.
In this topic, we will study the problems based on comparing values.
We will deal with few signs.
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In the examination, you will be given two-equation. You will need to solve both the equation and answer the question.
Variables: Variables are the values that do not remain fixed. Variables are denoted by the English alphabet.
Example: a, b, c, d, r, x, y, z, etc.
Constant: Constant are the values that remain fixed.
Example: 1, 2, 3, 4 so on.
In banking exams mainly two types of questions come.
1) Liner Equation
2) Quadratic Equation
(A) Linear Equation:
a) Linear equation with one variable:
Example:
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Example:
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From this two-equation, we can clearly say that x > y.
b) Linear equation in two variables:
Example:
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Multiplying equation (i) with 3 and multiplying equation (ii) with 2 and then subtracting them we get,
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Putting the value of x in equation (i) we get y = 5
Therefore we can say that x < y.
(B) Quadratic Equation:
A quadratic equation is a second-order polynomial equation in a single variable. So quadratic equation has two solutions. These solutions may be both real or complex.
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Discriminant Rules:
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In the exam, you will be given two quadratic equations. You have to solve and find both the roots of that equation then need to make a relation.
Suppose both the equations are:
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Now, when we will solve this equation we will find six types of combinations.
Case | Roots of x/y | Roots of x/y |
(1) | ++ | ++ |
(2) | ++ | +- |
(3) | ++ | |
(4) | + - | - - |
(5) | + - | + - |
(6) | - - | - - |
Case (1): When roots of both the equations are positive.
Example 1:
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We can clearly see that both the values of x are less than both the values of y.
So, we can easily conclude that x > y.
Example 2:
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Now, If we put this number on the number lines:
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From this, we can conclude that x ≤ y.
Note: When the sign of bx is negative (-) and the sign of c is positive (+) then we will get both the roots positive. And in this case, we need to go into detail.
Case (2): When the roots of the one equation are both positive and the roots of the other equation are one negative and one positive.
Example:
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Now, putting the roots on the number lines:
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From this, we can conclude that relationships can not be established.
Case (3): When one equation has both roots positive (+) and another equation has both negative (-) roots.
Example:
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Now, putting the roots on the number lines:
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Now we clearly see that x > y.
Case (4): When the roots of one equation have one positive and one negative and roots of the other equation have both negative.
Example:
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Now, putting the value on the number lines:
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We can say that relation between x and y can not be established.
Case (5): When both the equations have one positive and one negative root.
Example:
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Now, putting the values on the number lines:
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So, we can clearly say that relation between x and y can not be established.
Note:
Whenever in a question the value of c is negative (-) in both the equation then the answer will be always CAN NOT BE DETERMINED.
Case (6): When the roots of both equations have both negative roots.
Example:
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Now, putting the values on the number lines:
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Now we can clearly say that x > y.
