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.

representation of the solution set of an inequation



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.

inequation 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 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}

solution set of an inequation



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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