Solving Trinomial Inequalities

Introduction :
An algebraic expression is a combination of constants and variables.  An algebraic expression containing three terms is known as trinomial.

Inequality is a relationship between two expressions that are not equal, often written in th form of an equation but with symbols '<' or '>' meaning 'is greater than ' or  'is less than'.

Trinominal inequality is an algebraic expression where the two expressions are not equal and the symbols used is '<' or ' >'.

Steps for solving trinominal inequalities:

Bring all the terms on one side of the inequality so that we get a trinomial.
Find the factors of the trinominal by using algebraic identities or by splitting middle method.
Apply the inequality to all the factors of the trinomial so obtained.

Examples of Trinominal Inequality - I

1) Solve x^2 +1>2x

Solution:  x^2 + 1 > 2x

subtract 2x on both sides

x^2 - 2x + 1 > 2x - 2x

x^2 - 2x + 1  > 0

we know, (a-b) 2 = a^2 - 2ab + b^2

plug in a = x and b = 1, we get,

x^2 - 2x + 1 = ( x -1)2

Thus,  x^2 - 2x + 1> 0

(x-1) 2 > 0

x - 1 > 0

Add 1 on both sides

x -1 + 1> 0 +1

x > 1

2)Solve the inequality  x^2 + 5x < 6

Solution:  x^2 + 5x < 6

Subtract 6 on both sides

x 2 + 5x - 6 < 6 -6

x 2 + 5x - 6 < 0

Now, we will find the factors of the trinomial,

x 2 + 5x - 6

Clearly the last term of the trinomial is not a perfect square.

We apply splitting the middle term method,

Coefficient of first term = 1,    Last term = 6

Least common factor of first term and last term = 6

Factors of 6 are,

2 x 3 = 6     and    6 x 1 = 6

We take the factors,

6 x 1 = 6   because  6-1 = 5 = middle term

Therefore,   x 2 + 5x - 6  =   x 2 + 6x -x -6

=  x(x+6) -1(x+6)

= (x-1)(x+6)

Therefore,     x^2 +5x -6 < 0

(x-1)(x+6) <0

(x-1) <0                 or               (x+5) < 0

x -1+1< 0+1          or               x +5 -5 < 0 -5

x < 1                        or              x < (-5)

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Solving Word Problems of Trinomial Inequalities
Solving word problems of trinomial inequalities means we first convert the word problem to mathematical form and then follow the steps to solve the inequality.

1) Square of a number is greater than the sum 4 and thrice the number. Find the number.

Solution:  Let the number be x

Square of a number = x^2

Sum of 4 and thrice the number = 4 + 3x

Therefore the inequality formed is,

x^2 > 4 + 3x

subtract  (4+3x) on both sides

x^2 -(4 + 3x) > 4 + 3x - ( 4+3x)

x^2 -4 -3x > 0

x^2 -3x -4 > 0

Now consider the trinomial

x^2 -3x - 4

First term and last term of the trinomial are perfect squares

Here, we will solve the trinominal by splitting the middle term because

middle term `!=` 2 * first term * last term.

Now,  coefficient of first term = 1

Coefficient of first term * last term = 1*4 = 4

Factors of 4 are,

2 x 2 = 4           and              4 x 1 = 4

we will take, 4 x 1 = 4     because -4 +1 = -3 = middle term

Therefore,  x^2 - 3x - 4 = x^2 -4x + x - 4

=  x(x-4)+1(x-4)

=  (x-4)(x+1)

Therefore,    x^2 -3x -4 > 0

(x-4)(x+1) >0

(x-4) > 0                   or          (x+1) > 0

x-4+4 > 0 + 4          or         x +1-1> 0-1

x > 4                           or         x > -1

The number is either 4 or (-1)