__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

For example,

2) The sign of inequality is ** unchanged**, if we

For example,

3) The sign of inequality is ** reversed** if we

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.**

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