Shortcuts in Quantitative Aptitude for Competitive Exams – Algebraic Expression & Inequalities
Algebraic Expression & Inequalities
An unknown quantity used in any equation may be constant known as variable. Variables are generally denoted by the last English alphabet x, y, z etc.
An equation is a statement of equality of two algebraic expressions, which involve one or more variables.
An equation in which the highest power of variables is one, is called a linear equation. These equations are called linear because the graph of such equations on the x-y cartesian plane is a straight line.
Linear Equation in one variable
A linear equation which contains only one variable is called linear equation in one variable.
The general form of such equations is ax + b = c, where a, b and c are constants and a * 0.
All the values of x which satisfy this equation are called its solution(s).
NOTE : An equation satisfied by all values of the variable is called an identity. For example : 2x + x = 3x.
Linear equation in two variables
General equation of a linear equation in two variables is ax + by + c = 0, where a, b * 0 and c is a constant, and x and y are the two variables. The sets of values of x and y satisfying any equation are called its solution(s).
Consider the equation 2x + y=4. Now, if we substitute x = -2 in the equation, we obtain 2.(-2) + y=4 or -4 + y = 4 or y=8.Hence(- 2,8) is a solution. If we substitute x = 3 in the equation, we obtain 2.3 + y = 4 or 6 + y = 4 or y=-2
Hence (3, -2) is a solution. The following table lists six possible values for x and the corresponding values for y, i.e. six solutions of the equation.
Systems of Linear equation
Consistent System : A system (of 2 or 3 or more equations taken together) of linear equations is said to be consistent, if it has at least one solution.
Inconsistent System: A system of simultaneous linear equations is said to be inconsistent, if it has no solutions at all.
e.g. X +Y = 9; 3X + 3Y = 8
Clearly there are no values of X & Y which simultaneously satisfy the given equations. So the system is inconsistent.
An equation of the degree two of one variable is called quadratic equation.
General form : ax2+bx+c=0……… ( 1 ) where a, b and c are all real number
and a ≠ 0.
For Example :
2x2-5x + 3 = 0; 2x2 – 5 = 0; x2+ 3x=0
If b2– 4ac > 0, then the quadratic equation gives two and only two values (either same or different) of the unlaiown variable and both these values are called the roots of the equation.
The roots of the quadratic equation (1) can be evaluated using the following formula.
The above formula provides both the roots of the quadratic equation, which are generally denoted by α and β ,
The expression inside the square root b -4ac is called the DISCRIMINANT of the quadratic equation and denoted byD. Thus, Discriminant (D) = b2– 4ac.
Nature of Roots
The nature of roots of the equation depends upon the nature of its discriminant D.
If D < 0, then the roots are non-real complex, Such roots are always conjugate to one another. That is, if one root is p + iq then other is p – iq, q ≠ 0. If D = 0, then the roots are real and equal. Each root of the equation becomes -b/2a. Equal roots are referred as repeated roots or double roots also. If D > 0 then the roots are real and unequal.
Sign of Roots:
Let α , β. are real roots of the quadratic equation ax2 + bx + c = 0 that is D = b2– 4ac ≥ 0 . Then.
- Both the roots are positive if a and c have the same sign and the sign of b is opposite.
- Both the roots are negative if a, b and c all have the same sign.
- The Roots have opposite sign if sign of a and c are opposite.
- The Roots are equal in magnitude and opposite in sign if b = 0 [that is its roots a and -a]
- The roots are reciprocal if a = c.
[that is the roots are a and 1/α ]
Symmetric Functions of Roots :
An expression in α β is called a symmetric function of α β . if the function is not affected by interchanging α and β . if α β are the roots of the quadratic equation ax2+ bx + c = 0, a ≠ 0 then,
Formation of quadratic Equation with Given Roots:
A statement or equation which states that one thing is not equal to another, is called an inequation.
- ‘ < ’ means “is less than”
- ‘ >’ means “is greater than”
- ‘ ≤ ’ means “is less than or equal to”
- ‘ ≥ ’ means “is greater than or equal to”
For example :(a) x < 3 me ans x is less than 3.
(b) y ≥ 9 means y is greater than or equal to 9.
- Adding the same number to each side of an inequation does not effect the sign of inequality, i.e. if x > y then, x + a > y + a.
- Subtracting the same number to each side of an inequation does not effect the sign of inequaltiy, i.e.,if x < y then, x-a<y-a.
- Multiplying each side of an inequality with same positive number does not effect the sign of inequality, i.e., if x ≤ y then ax ≤ ay (where, a >0).
- Multiplying each side of an inequality with a negative number reverse the sign of inequality i.e., if x < y then ax > ay (where a <0).
- Dividing each side of an inequation by a positive number does not effect the sign of inequality, i.e., if x ≤ y then ≤ (where a > 0).
- Dividing each side of an inequation by a negative number reverses the sign of inequality, i.e., if x > y then < (where a < 0).
If the signs of all the terms of an inequality are changed, then the sign of the inequality will also be reversed.
Applications Formulation of Equations/ Expressions:
A formula is an equation, which represents the relations between two or more quantities.
For example :
Area of parallelogram (A) is equal to the product of its base (b) and height (h), which is given by A = b x h or A = bh.
Perimeter of triangle (P),
P = a + b + c, where a, b and c are length of three sides.
More Applications of Equations :
Problems on Ages can be solved by linear equations in one variable, linear equations in two variables, and quadratic equations.
- If present age of the father is F times the age of his son. T years hence, the father’s age become Z times the age of son then present age of his son is given by
- If T1 years earlier the age of the father was n times the age of his son, T2 years hence, the age of the father becomes m times the age of his son then his son’s age is given by
- Present age of Father : Son = a : b After / Before T years = m : n