Chapter-6

Linear Inequalities

Inequalities and Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation:

What is inequalities

  • In mathematics, an inequality is a relation that holds between two values when they are different
  • Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative 
  • Two real numbers or two algebraic expressions related by the symbol ‘<’, ‘>’, ‘£’ or ‘³ form an inequality.

Symbols used in inequalities

  • The symbol < means less than. The symbol > means greater than.
  • The symbol < with a bar underneath means less than or equal to. Usually this is written as 
  • The symbol > with a bar underneath means greater than or equal to. Usually this is written as 
  • The symbol  means the quantities on left and right side are not equal

Examples

  • a < b means a is less then b or b is greater a
  • a≤b means a is less then or equal to b
  • a > b means a is greater than b
  • a≥b means a is greater or equal to b

Types of inequalities:

  1. Numerical inequalities :

Example:

3 < 5(read as 3 is less than 5)

7 > 5 (read as 7 is greater than 5)

  1. Literal inequalities:

Inequalities which involve variables are called literal inequalities

Example: x < 5; y > 2

    x ³5; y £ 2;

  1. Double inequalities:

The inequality is said to be a double inequality if the statement shows the double relation of the expressions or the numbers..

Example: 3 < 5 < 7 (read as 5 is greater than 3 and less than 7),

3 £ x < 5 (read as x is greater than or equal to 3 and less than 5)

And  2 < y £ 4(read as y is less than or equal to 4 and greater than 2)

  1. Open Sentence :

The inequality is said to be an open sentence if it has only one variable.

Example: x < 6 (x is less than 6),

 x > 8

  1. Strict inequalities:

A relation that expresses the comparison between the unequal quantities is called strict inequality.

Example: ax + b < 0,

ax + b > 0,

ax + by < c,

ax + by > c,

ax2 + bx + c > 0

  1. Slack inequalities:

 Inequalities involving the symbol '≥' or '≤' are called slack inequalities.

Example: ax + by +c,

ax + by +c,

3x – y ≥ 5

  1. Linear inequalities:

When two expressions are connected by “greater than” or “less than” sign, we get an inequality. Also, the linear inequation is similar to a linear equation, where the equal to sign is replaced by the inequality sign.

Example: ax + b < 0,

ax + b > 0,

 x - 5 > 3x – 10,

  1. Quadratic inequalities:

A quadratic inequality is an inequality that contains a quadratic expression.

Example:  ax2+bx+c<0,

ax2 + bx + c > 0

ax2 + bx + c £ 0

 

Things are which are safe to do in inequality which does not change in direction:

  • Addition of same number on both sides
            a>b
    =>    a+c > b+c
  • Subtraction of same number on both sides
          a>b
    =>  a−c > b−c
  • Multiplication/Division by same positive number on both sides
        a>b
    if c is positive number then
      ac >  bc
    if c is positive number (non zero) then
      a/c >  b/c

Things which changes the direction of the inequality:

  • Swapping the left and right sides
  • Multiplication/Division by negative number on both sides

If a > b and  c is negative number then  ac <  bc

If a < b and  c is negative number then  ac > bc

  • Don’t multiple by variable whose values you dont know as you don’t know the nature of the variable

Concept Of Number line:

  • A number line is a horizontal line that has points which correspond to numbers. The points are spaced according to the value of the number they correspond to; in a number line containing only whole numbers or integers, the points are equally spaced.

  • It is very useful in solving problem related to inequalities and also representing it
    Suppose x >2(1/ 3), this can represent this on number line like that

Linear Inequation in One Variable:

A inequation of the form
ax+b>0
or
ax+b≥0

or
ax+b<0
or
ax+b≤0
are called the linear equation in One Variable

      Example:

      x−2<0,

          3x+10>0,

10x−17≥0

Linear Inequations in Two Variable:

A inequation of the form
ax+by>c
or
ax+by≥c
or
ax+by<c
or
ax+by≤c
are called the linear equation in two Variable

Example:
 

  • x−2y<0
  • 3x+10y>0
  • 10x−17y≥0

How to solve the linear inequalities in one variable?:

Steps to solve the Linear inequalities in one variable:

  • Obtain the linear inequation
  • Pull all the terms having variable on one side and all the constant term on another side of the inequation
  • Simplify the equation in the form given above
    ax>b
    or
    ax≥b
    or
    ax<b
    or ax≤b
  • Divide the coefficent of the variable on the both side.If the coefficent is positive,direction of the inequality does not changes,but if it is negative, direction of the inequation changed
  • Put the result of this equation on number line and get the solution set in interval form

Example:
x−2 > 2x+15
Solution
x−2 > 2x+15
Subtract x from both the side x−2−x > 2x+15−x
−2>x+15

Subtract 15 from both the sides −17 > x
Solution set (−∞,−17)

Example: Find all pairs of consecutive odd natural numbers, both of which are larger

than 10, such that their sum is less than 40.

Solution: Let x be the smaller of the two consecutive odd natural number, so that the

other one is x +2. Then, we should have

x > 10 ... (1)

and x + ( x + 2) < 40 ... (2)

Solving (2), we get

2x + 2 < 40

i.e., x < 19 ... (3)

From (1) and (3), we get

10 < x < 19

Since x is an odd number, x can take the values 11, 13, 15, and 17. So, the required

possible pairs will be

(11, 13), (13, 15), (15, 17), (17, 19)


Steps to solve the linear inequality of the fraction form

[(ax+b)/(cx+d)]>k or similar type
 

  • Take k on the LHS
  • Simplify LHS to obtain the inequation in the form
    [(px+q)/(ex+f)]>0
    Make the coefficent positive if not
  • Find out the end  points solving the equatoon px+q=0 and ex+f=0
  • Plot these numbers on the Number line. This divide the number into three segment
  • Start from LHS side of the number and Substitute some value in the equation in all the three segments to find out which segments satisfy the equation
  • Write down the solution set in interval form

Or there is more method to solves these
 

  • Take k on the LHS
  • Simplify LHS to obtain the inequation in the form

[(px+q)/(ex+f)]>0
Make the coefficent  positive if not

  •   For the equation to satisfy both the numerator and denominator must have the same sign
  • So taking both the part +, find out the variable x interval
  • So taking both the part -, find out the variable x interval
  • Write down the solution set in interval form

Lets take one example to clarify the points

Lets take one example to clarify the points

Question

1. Solve     X−3x+5>0
Solution:
Method A

 

  1. Lets find the end points of the equation
    Here it is clearly
    x=3 and x=-5
  2. Now plots them on the Number line
  3. Now lets start from left part of the most left number
    i.e
    Case 1

    x<−5 ,Let takes x=-6 then −6−3−6+5>0
    3>0
    So it is good
    Case 2
    Now take x=-5
    as x+5 becomes zero and we cannot have zero in denominator,it is not the solution
    Case 3
    Now x > -5 and x < 3, lets take x=1 then 
    1−31+5>0
    −16>0
    Which is not true
    Case 4
    Now take x =3,then
    0> 0 ,So this is also not true
    case 5
    x> 3 ,Lets x=4

    4−34+5>0
    19>0
    So this is good
  4. So the solution is
    x < -5 or x > 3
    or

    (−∞,−5)∪(3,∞)

Method B
 

  1. the numerator and denominator must have the same sign. Therefore, either
    1) 
    x−3>0 and x+5>0,or 2) x - 3 < 0 and x + 5 < 0$
  2. Now, 1) implies x > 3 and x > -5.
    Which numbers are these that are both greater than 3 and greater than -5?
    Clearly, any number greater than 3 will also be greater than -5. Therefore, 1) has the solution
    x > 3.
  3. Next, 2) implies
    x < 3 and x < -5.
    Which numbers are these that are both less than 3 and less than -5?
    Clearly, any number less than -5 will also be less than 3. Therefore, 2) has the solution
    x < -5.
  4. The solution, therefore, is
    x < -5 or x > 3

 

Quadratic Inequality:

A inequation of the form
 so for a  0
ax2+bx+c>0
or
ax2+bx+c<0
or
ax2+bx+c≥0
or
ax2+bx+c≤0
are called Quadratic Inequation in one Variable

Steps to solve Quadratic or polynomial inequalities:

ax2+bx+c>0,
or
ax2+bx+c<0,
or
ax2+bx+c≥0,
ax2+bx+c≤0

  1. Obtain the Quadratic inequation
  2. Pull all the terms having on one side and Simplify the equation in the form given above
  3. Find the roots(0 points) of the Quadratic equation using any of the method and write in this form
    (x-a)(x-b)
  4. Plot these roots on the number line .This divide the number into three segment
  5. Start from LHS side of the number and Substitute some value in the equation in all the three segments to find out which segments satisfy the equation
  6. Write down the solution set in interval form

Solved Example
Question 1
x2-5x+6>0
Solution
1) Simplify or factorize the inequality which means factorizing the equation in case of quadratic equalities
Which can be simplified as

x2−5x+6>0
=> x2−2x−3x+6>0
=> (x−2)(x−3)>0
2) Now plot those points on Number line clearly
3) Now start from left of most left point on the Number line and look out the if inequalities looks good or not. Check for greater ,less than and equalities at all the end points
So in above case of 
x2−5x+6>0
We have two ends points 2 ,3
Case 1
So for 
x<2 ,Let take x=1,then (1−2)(1−3)>0
=> 2>0
So it is good
So This inequalities is good for x < 2
Case 2
Now for x =2,it makes it zero, so not true. Now takes the case of 
2<x<3. Lets takes x= 2.5
(2.5−2)(2.5−3)>0
=> −0.25>0
Which is not true so this solution is not good
Case 3
Now lets take the right most part i.e 
x>3
Lets take x=4
(4−2)(4−3)>0
=> 2>0
So it is good.
Now the solution can either be represented on number line or we can say like this

(−∞,2)(3,∞)

Question: Solve y < 2 graphically

Graph of y = 2 is given in the Figure.

Let us select a point, (0, 0) in lower half plane I and putting y = 0 in the given inequality, we see that

1 × 0 < 2 or 0 < 2 which is true.

Thus, the solution region is the shaded region below the line y = 2.

 Hence, every point below the line (excluding all the points on the line) determines the solution of the given inequality.