GRAPH INEQUALITIES ON A NUMBER LINE

1. If we have the inequality

< (less than) or > (greater than),

we have to use the empty / unfilled circle.

2. If we have the inequality sign 

≤ (less than or equal to) or ≥ (greater than or equal to),

we have to use the filled circle.

Example 1 :

Solve the following linear inequality and graph.

2x - 4 ≤ 0

Solution :

2x - 4 ≤ 0

Add 4 on both sides

2x - 4 + 4 ≤ 0 + 4

2x ≤ 4

Divide by 2 on both sides

2x/2 ≤ 4/2

x ≤ 2

So, any real number less than or equal to 2 is a solution of the given equation.

The solution set of the given inequality is (-∞, 2].

Example 2 :

Solve the following linear inequality and graph.

-3x + 12 < 0

Solution :

-3x + 12 < 0

Subtract 12 on both sides

-3x + 12 - 12 < 0 - 12

-3x < -12

Divide by -4 on both sides

-3x/(-3) < -12/(-3)

x < 4

So, any real number less 4 is a solution of the given equation.

The solution set of the given inequality is (-∞, 2].

Example 3 :

Solve the following linear inequality and graph.

4x - 12 ≥  0

Solution :

4x - 12 ≥  0

Add 12 on both sides

4x - 12 + 12  ≥  0 + 12

4x  ≥  12

Divide by 4 on both sides

4x/4 ≥ 12/4

x ≥ 3

So, any real number greater than or equal to 3 is a solution of the given equation.

The solution set of the given inequality is [3, ∞).

Example 4 :

Solve the following linear inequality and graph.

7x + 9 > 30

Solution :

7x + 9 > 30

Subtract 9 on both sides

7x + 9 - 9 > 30 - 9

7x  > 21

Divide by 7 on both sides

7x/7 > 21/7

x > 3

So, any real number greater than 3 is a solution of the given equation.

The solution set of the given inequality is (3, ∞).

Example 5 :

Solve the following linear inequality and graph.

5x - 3 < 3x + 1

Solution :

5x - 3 < 3x + 1

Subtract 3x on both sides

5x - 3 - 3x  < 3x + 1 - 3x 

2x - 3 <  1 

Add 3 on both sides

2x - 3 + 3 <  1 + 3

2x < 4

Divide by 2 on both sides

2x/2 < 4/2

x < 2

So, any real number lesser than 2 is a solution of the given equation.

The solution set of the given inequality is (2, ∞).

Example 6 :

Solve the following linear inequality and graph.

3x + 17 ≤ 2(1 - x)

Solution :

3x + 17 ≤ 2(1 - x)

3x + 17 ≤ 2 - 2x

Add 2x on both sides

3x + 2x + 17  ≤ 2 - 2x + 2x

5x + 17 ≤ 2 

Subtract 17 on both sides

5x + 17 - 17 ≤ 2 - 17

5x ≤ -15

Divide by 5 on both sides

5x/5 ≤ -15/5

x ≤ -3

So, any real number lesser than or equal to -3 is a solution of the given equation.

The solution set of the given inequality is (-∞ , -3].

Example 7 :

Solve the following linear inequality and graph.

2(2x + 3) - 10 ≤ 6 (x - 2)

Solution :

2(2x + 3) - 10 ≤ 6 (x - 2)

4x + 6 - 10 ≤ 6 x - 12

4x - 4 ≤ 6 x - 12

Subtract 6x on both sides

4x - 4 - 6x  ≤ 6 x - 12 - 6x 

-2x - 4 ≤ - 12

Add 4 on both sides

-2x - 4 + 4 ≤ - 12 + 4

-2x ≤ - 8

Divide by -2 on both sides

-2x / (-2) ≤ - 8 / (-2)

x ≤ 4

So, any real number lesser than or equal to 4 is a solution of the given equation.

The solution set of the given inequality is (-∞ , 4].

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