Linear Inequalities

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)

\( \scriptsize 3x \: – \: 5 \: + \: 5 > \: -11 \: + \: 5 \)

\( \scriptsize 3x > \: -6 \)

(dividing through by +3)

\( \frac{3x}{3} > \: \frac{-6}{3} \)

\( \scriptsize x > \: -2 \)

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. \( \scriptsize -3 < 5\: – \: 3x\)

\( \scriptsize 3x < 5\: + \: 3\)

\( \scriptsize 3x < 8\)

\( \scriptsize x < \normalsize \frac{8}{3}\)

\( \scriptsize x < 2 \frac{2}{3}\)

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

\( \scriptsize – 3x \leq 11 \: – \: 5\)

\( \scriptsize – 3x \leq 6\)

(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