A linear inequality is similar to a linear equation, but instead of an equals sign it uses an inequality symbol. Linear inequalities can involve whole number or fractional coefficients and are solved by isolating the variable.

 

The inequality symbols used are:
\( < \) less than
\( > \) greater than
\( \le \) less than or equal to
\( \ge \) greater than or equal to

 

 

Forming Linear Inequalities

Inequalities can be formed from worded statements.

 

For example, the statement
“Three times a number is greater than 12”
can be written as:

$$
3x > 12
$$

 

The statement
“A number divided by 4 is less than or equal to 5”
can be written as:

$$
\frac{x}{4} \le 5
$$

 

 

Solving Linear Inequalities with Whole Number Coefficients

The method for solving inequalities is the same as for equations, except when multiplying or dividing by a negative number.

 

For example:

$$
3x > 12
$$

 

Divide both sides by \( 3 \):

$$
x > 4
$$

 

Another example:

$$
7x - 5 \le 16
$$

 

Add \( 5 \) to both sides:

$$
7x \le 21
$$

 

Divide by \( 7 \):

$$
x \le 3
$$

 

 

Solving Linear Inequalities with Fractional Coefficients

Fractional coefficients can be handled by using inverse operations or by clearing fractions.

 

For example:

$$
\frac{x}{3} > 2
$$

 

Multiply both sides by \( 3 \):

$$
x > 6
$$

 

Another example:

$$
\frac{2x}{5} \le 4
$$

 

Multiply both sides by \( 5 \):

$$
2x \le 20
$$

 

Divide by \( 2 \):

$$
x \le 10
$$

 

 

Inequalities Involving Negative Coefficients

When multiplying or dividing by a negative number, the inequality sign must be reversed.

 

For example:

$$
-2x > 6
$$

 

Divide both sides by \( -2 \) and reverse the inequality sign:

$$
x < -3
$$

 

Another example:

$$
- \frac{x}{4} \le 5
$$

Multiply both sides by \( -4 \) and reverse the inequality sign:

$$
x \ge -20
$$

 

 

Writing the Final Answer

Solutions to inequalities are often written as a statement showing the range of values that the variable can take.

 

For example:

$$
x \ge 2
$$

This means all values of \( x \) greater than or equal to 2 satisfy the inequality.

 

 

Key Points to Remember

Solve inequalities using inverse operations, just like equations.
Reverse the inequality sign when multiplying or dividing by a negative number.
Work carefully with fractions and signs.

 

Being confident with linear inequalities is important for solving problems involving limits, constraints and ranges of values.