SOLVING LINEAR INEQUALITIES

Example 1 :

Solve 23x < 100 when

(i) x is a natural number,  (ii) x is an integer.

Solution :

In order to satisfy, let us apply some random values of x.

The set of values for which the above inequality satisfy is the solution.

23x < 100 

x  =  5

23(5) < 100

115 < 100

Does not satisfy

(i) x is a natural number

Hence the required solution is

{1, 2, 3, 4} (natural number starts with 1).

(ii) x is an integer

{-∞, ...........0, 1, 2, 3, 4}

Example 2 :

Solve −2x ≥ 9 when (i) x is a real number, (ii) x is an integer, (iii) x is a natural number.

Solution :

First, let us solve for x

−2x ≥ 9

Divide both sides by -2

x ≤ -9/2

(i) x is a real number

The above inequality satisfies for every negative values of x upto -9/2.

Hence the required solution is (-∞, -9/2]

(ii) x is an integer

Since the value of x must be integer, we could not choose a rational number for x.

Hence the required solution is -∞, .............,-7, -6, -5 

(iii) x is a natural number

Natural number starts with 1, so we have to choose only positive numbers.

So, there is no solution.

Example 3 :

Solve for x

Solution :

(i)    9 (x - 2) ≤ 25 (2 - x)

9x - 18 ≤ 50 - 25x

Add both sides by 25x.

9x + 25x - 18 ≤ 50

34x - 18 ≤ 50

Add both sides by 18

34x ≤ 50 + 18

34x ≤ 68

Divide both sides by 34

x ≤ 68/34

x ≤ 2

So, the required solution is (-∞, 2].

Solution :

(5 - x)/3 < (x - 8)/2

2(5 - x) < 3(x - 8)

10 - 2x < 3x - 24

Subtract both sides by 3x

10 - 5x < -24

Subtract both sides by 10

-5x < -24 - 10

-5x < -34

x > 34/5

So, the required solution is [34/5, ∞).

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.