SOLVING AND GRAPHING INEQUALITIES

If an inequality is in the form

(x < a)  or  (x > a)  or  (x ≤ a)  or  (x ≥ a),

where a is a constant, we can easily sketch the graph of the inequality.

In case, the inequality is in the form

ax + b < bx + d,

where a, b and d are constants, we have to solve for x and sketch the graph.

Solving Inequalities

Step 1 :

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

Step 2 :

Both sides of an inequality can be multiplied or divided by 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 a negative number.

Step 3 :

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

Graphing Inequalities

In graphing inequalities in one variable on a number line, we have to follow the steps given below.

Step 1 :

If we have one of the signs like < (less than) or > (greater than), we have to use the unfilled circle.

Step 2 :

If we have one of the signs like ≤ (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].

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.