Form 2. Unit 7: Lesson 3- Solving Inequality

Objectives

At the end of the lesson, students should be able to solve:

• An Inequality with the variable on one side.
• An Inequality with the variable on both sides.
• An Inequality containing brackets.
• An Inequality involving fractions.

Inequality questions are solved just like equality questions. The only difference is when dividing or multiplying by a NEGATIVE number. The sign of inequality is REVERSED if we multiply or divide both sides of the inequality by a negative number.

Rules for solving inequality

1) The same number may be added or subtracted from both sides of an inequality without changing the inequality.

For example,

2) The sign of inequality is unchanged, if we multiply or divide both sides by a positive number.

For example,

3) The sign of inequality is reversed if we multiply or divide both sides of an inequality by a negative number.

i.e. > changes to <

≤ changes to ≥ and vice versa.

For example,

Note that if we leave the sign unchanged, the inequality (–4 > –3) will be read as – 4 is greater than –3 and (–2 < –3) will be read as –2 is less than –3 which is not true. Hence, we change the sign to make the inequality true; –4 < –3.

Inequality with the variable on one side

Example 1

Find the truth set of 4x ≤ 12

Solution

Divide both side by 4.

 

 

Hence the truth set of the above inequality is {x: x ≤ 3}

Example 2

Find the truth set of – 4x ≤ 40

Solution

Divide both side by –4.

 

 

 

Hence the truth set of the above inequality is {x: x ≥ –10}.

Note

The sign has been reversed because we divided both sides of the inequality by a negative number (i.e. – 4).

 Inequality with the variable on both sides

Example 1

Find the truth set of  8x + 1 < 4x – 3

Solution

Group like terms.

Simplify both side.

Divide both side by 4.

 

 

The truth set of solution set of the above inequality is {x: x < -1}.

Example 2

Find the truth set of 9x + 4 ≥ 5x + 6, illustrate your answer on a number line.

Solution

Group like terms.

Simplify both side.

Divide both side by 4.

 

Inequality containing brackets

If an inequality contains brackets, expand to remove the brackets.

Take extra care when there is a ‘–’   sign outside the bracket.

Example 1

Find the truth set of 2(x + 11) +3x ≤ 42.

Solution

Expand  to remove the bracket

Group like terms.

Simplify both sides.

Divide both side by 6.

 

{x:x≤3.33}

Example 2

Find the truth set of 2(2x 1) + 4 > 5(x + 4) + 2

Solution

Expand  to remove the bracket.

Group like terms.

Simplify both side.

Divide both side by –1.

 

Inequality involving fractions

When an inequality involves fractions, remove fractions by multiplying each term of the inequality by the L.C.M of the denominators of the fractions.

Example 1

Find the range of the values of x for which

 

Solution

Multiply both terms by 12(i.e. the L.C.M). 

Expand  to remove the bracket.

Group like terms.

Simplify both sides.

 

Example 2

Find the truth set of the following inequalities and illustrate your answer on a number line.

Solution

Multiply both terms by 10 (i.e. the L.C.M).             

  Expand to remove the bracket.           

 Group like terms.

Divide both side by -7

Hence the solution set of the above inequality is  

The solution set can be represented on the number liner as show below

Hence the solution set of the above inequality is {x: x ≤ 8}

The solution set can be represented on the number liner as show below.

Example 3

Find the truth set of

where x is a real number. Illustrate your answer on a number line.

Solution

                                                  

                                                 

 

 

 

Hence the solution set of the above inequality is {x: x ≤ 14}.

The solution set can be represented on the number liner as shown below.

General step for solving an Inequality

Step1:

Check if there are fractions.

If there are fractions multiply both sides of the inequality by the LCM of the denominator to clear fractions.

Note

If there are mixed fractions, first change them to improper fractions. 

Step 2

Check if there are brackets.

If there are brackets expand to remove the brackets.

Step 3

Collect or group all like terms on one side.

Usually we group all terms containing the variable on the LHS of the inequality.

This step is the same as adding or subtracting the same quantity from both sides of the inequality to get rid of an unwanted quantity.

Note

If a positive term crosses the inequality sign it becomes negative and if a negative crosses the equality sign it become positive.