Inequality is a mathematical statement that compares two expressions. The statement resembles the usual equation except that the equality sign (=) is replaced by one of the following inequality signs:

\( \scriptsize < \) is less than

\( \scriptsize > \) is greater than

\( \scriptsize \leq \) is less than or equal to

\( \scriptsize \geq \) is greater than or equal to

The rules for dealing with inequalities are:**1.** We can add the same number to both sides.**2.** We can subtract the same number from both sides of the inequality.**3.** We multiply or divide both sides of the inequality by the same positive number.**4. **We can multiply or divide both sides of the inequality by the same negative number if, and **only if**, we change the inequality sign i.e > to <, < to >, ≤ to ≥, ≥ to ≤.

Also note that solutions of inequalities are always a range of values.

### Example 1

1. Solve \( \scriptsize 3x \: – \: 5 > \: -11 \)

**(add 5 to both sides)**

**(dividing through by +3)**

The circle at – 2 is not shaded and it means that – 2 is not included

### Example 2

Solve \( \scriptsize -3 < 5\: – \: 3x \: \leq 11\)

Take each inequality separately

**i.** \( \scriptsize -3 < 5\: – \: 3x\)

**ii.** \( \scriptsize 5 \: – \: 3x \leq 11\)

(dividing by – 3)

\( \frac {-3x}{-3} \geq \frac {6}{-3} \) \( \scriptsize x \geq -2\)The solutions are combined as \( \scriptsize -2 \leq x < 2 \frac{2}{3} \)

This is shown on a number line as

The shaded circle at -2 means -2 is included.

x lies in an interval which is closed at -2 and open at \( 2 \frac{2}{3} \)

### Example 3

Solve \( \frac{4x \: – \: 1}{3} \: – \: \frac{1 \: + \: 2x}{5}\scriptsize \leq 8 \: + \: 2x \)

Multiply through by the L.C.M which is 15

\( \scriptsize 5(4x \: – \: 1) \: – \: 3(1 \: + \: 2x) \leq 15(8 \: + \: 2x) \) \( \scriptsize 20x \: – \: 5 \: – \: 3 \: – \: 6x \leq 120 \: + \: 30x \) \( \scriptsize 20x \: – \: 6x \: – \: 30x \leq 120 \: + \: 5 \: + \: 3 \) \( \scriptsize -16x \leq 128 \) \( \scriptsize x \geq \: – \normalsize \frac{128}{16} \) \( \scriptsize x \geq \: – 8 \)This is shown on a number line as

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