SOLVING LINEAR INEQUALITIES IN ONE VARIABLE EXAMPLES

Let a be a non zero real numbers and x be a variable.

Then, the inequality of the form

ax + b  <  0

ax + b  ≤  0

ax + b  >  0

ax + b  ≥  0

are known as linear inequalities in one variable.

The following rules will be useful to solve linear inequalities in one variable. 

Rule 1 : 

Same number may be added to (or subtracted from) both sides of an inequality without changing the sign of inequality.

Rule 2 : 

Both sides of an inequality can be multiplied (or divided) both by the same positive real number without changing the sign of inequality. However, the sign of inequality is revered when both sides of an inequality are multiplied or divided by the negative number.

Rule 3 :

Any term of an inequality may be taken to the other side with its sign changed without affecting sings of inequality.

Let us see some examples based on the above concept.

Example 1 :

Solve 5x - 3 < 3x + 1 when

(i) when x is a real number

(ii) when x is an integer

(iii) when x is a natural number

Solution :

(i) When x is a real number :

5x - 3  <  3x + 1

Subtract 3x from each side. 

2x - 3  <  1 

Add 3 to each side.

2x  <  4

Divide each side by 2

x  <  2

Because x is real number, the solution set is 

(-∞, 2)

(ii) When x is an integer :

We have already solved for x in the given inequality.

That is

x  <  2

Because x is an integer, the solution set is  

{...............,-4, -3, -2, - 1, 0, 1}

(iii) When x is a natural number : 

x  <  2

Because x is a natural number, the solution set is  

{ 1 }

Example 2 :

Solve for x : 

3x + 17  ≤  2(1 - x)

Solution :

3x + 17  ≤  2(1 - x)

3x + 17  ≤  2 - 2x

Add 2x to each side. 

5x + 17  ≤  2

Subtract 17 from each side.

5x  ≤  - 15

Divide each side by 5.

x  ≤  - 3

So, the solution set is

(-∞, -3]

Example 3 :

Solve for x : 

2(2x + 3) - 10  ≤  6(x - 2)

Solution :

 2(2x + 3) - 10  ≤  6(x - 2)

4x + 6 - 10  ≤  6x - 12

4x - 4  ≤  6x - 12

Subtract 6x from each side. 

-2x - 4  ≤  - 12

Add 4 to each side.

-2x  ≤  - 8

Divide each side by (-2). 

x  ≥  4

So, the solution set is 

[4, ∞)

Example 4 :

Solve for x : 

3x - 7  >  x + 1

Solution :

3x - 7  >  x + 1

Subtract x from each side. 

2x - 7  >  1

Add 7 to each side. 

2x  >  8

Divide each side by 2. 

x  >  4

So, the solution set is

(4, ∞)

Example 5 :

Solve for x : 

- (x - 3) + 4  <  5 - 2x

Solution :

- ( x - 3) + 4  <  5 - 2x

-x + 3 + 4  <  5 - 2x 

-x + 7  <  5 -2x

Add 2x to each side.

x + 7  <  5

Subtract 7 from each side. 

x  <  - 2

So, the solution set is (-∞, -2). 

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