Representation of the Solution Set of an Inequation

Graphical representation of the solution set of an inequation:

A number line is used to represent the solution set of an inequation graphically.

● First solve the linear inequation and find the solution set.

● Mark it on the number line by putting a dot.

● In case the solution set is infinite, then put three more dots to indicate infiniteness.

For Example: 

1. Solve the inequation 3x - 5 < 4, x ∈ N and represent the solution set graphically. 

Solution:

We have 3x - 5 < 4

⇒ 3x - 5 + 5 < 4 + 5 (Add 5 to both sides)

⇒ 3x < 9

⇒ 3x/3 < 9/3 (Divide both sides by 3)

⇒ x < 3

So, the replacement set = {1, 2, 3, 4, 5, ...}

Therefore, the solution set = {1, 2} or S = {x : x ∈ N, x < 3}

Let us mark the solution set graphically.

Solution set is marked on the number line by dots. 

2. Solve 2x + 8 ≥ 18 

Here x ∈. W represent the inequation graphically

⇒ 2x + 8 - 8 ≥ 18 - 8 (Subtract 8 from both sides)

⇒ 2x ≥ 10

⇒ 2x/2 ≥ 10/2 (Divide both sides by 2)

⇒ x ≥ 5

Replacement set = {0, 1, 2, 3, 4, 5, 6, ...}

Therefore, solution set = {5, 6, 7, 8, 9, ...}

or, S = {x : x ∈ W, x ≥ 5}

Let us mark the solution set graphically.

Solution set is marked on the number line by dots. We put three more dots indicate infiniteness of the solution set.

3. Solve -3 ≤ x ≤ 4, x ∈ I

Solution:

This contains two inequations,

-3 ≤ x and x ≤ 4

Replacement set = {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ...}

Solution set for the inequation -3 ≤ x is -3, -2, -1, 0, 1, 2, ... i.e., S = {-3, -2, -1, 0, 1, 2, 3, ...} = P

And the solution set for the inequation x ≤ 4 is 4, 3, 2, 1, 0, -1, ... i.e., S = {..., -3, -2, -1, 0, 1, 2, 3, 4} = Q

Therefore, solution set of the given inequation = P ∩ Q

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

or S = {x : x ∈ I, -3 ≤ x ≤ 4}

Let us represent the solution set graphically.

Solution set is marked on the number line by dots.

A number line is used for representation of the solution set of an inequation.

Now, solution set S = {3, 4, 5, 6, ...} S = (x : x ∈ N, x > 3)

For Example:

4. 2x + 3 ≤ 15

⇒ 2x + 3 - 3 ≤ 15 - 3 (Subtract 3 from both sides)

⇒ 2x ≤ 12 ⇒ 2x/2 ≤ 12/2 (Divide both sides by 2)

⇒ x ≤ 6

Now, the solution set S = {1, 2, 3, 4, 5}   S' = {x : x ∈ N, x < 6}

Now, S ∩ S’ = {3, 4, 5, 6}

5. 0 < 4x - 9 ≤ 5,     x ∈ R

Solution:

Case I: 0 ≤ 4x - 9

0 + 9 ≤ 4x - 9 + 9

⇒ 9 ≤ 4x

⇒ 9/4 ≤ 4x/4

⇒ 2.25 ≤ x

⇒ 2.2 < x

Case II: 4x - 3 ≤ 9

⇒ 4x - 3 + 3 ≤ 9 + 3

⇒ 4x ≤ 12

⇒ x ≤ 3

S ∩ S' = {2.2 < x ≤ 3} x ∈ R

           = {x : x ∈ R 3 ≥ x > 2.2}

Arrow on right shows that solution set continues.

● Inequations

What are Linear Inequality?

What are Linear Inequations?

Properties of Inequation or Inequalities

Representation of the Solution Set of an Inequation

Practice Test on Linear Inequation

● Inequations - Worksheets

Worksheet on Linear Inequations

7th Grade Math Problems

8th Grade Math Practice 

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