Linear inequalities can be more complex when the variable appears on both sides of the inequality or when two inequalities are combined into a double inequality. The aim is still to find all values of the variable that make the inequality true.

 

The same principles apply as for equations, with careful attention to inequality signs.

 

 

Linear Inequalities with the Variable on Both Sides

When the variable appears on both sides, the first step is to collect all variable terms on one side of the inequality.

 

For example:

$$
5x + 2 > 2x + 11
$$

 

Subtract \( 2x \) from both sides:

$$
3x + 2 > 11
$$

 

Subtract \( 2 \):

$$
3x > 9
$$

 

Divide by \( 3 \):

$$
x > 3
$$

 

Another example:

$$
7x - 4 \le 3x + 8
$$

 

Subtract \( 3x \) from both sides:

$$
4x - 4 \le 8
$$

 

Add \( 4 \):

$$
4x \le 12
$$

 

Divide by \( 4 \):

$$
x \le 3
$$

 

 

Inequalities Requiring the Sign to Be Reversed

If the final step involves dividing or multiplying by a negative number, the inequality sign must be reversed.

 

For example:

$$
2 - 4x > 10
$$

 

Subtract \( 2 \):

$$
-4x > 8
$$

 

Divide by \( -4 \) and reverse the sign:

$$
x < -2
$$

 

 

Double Inequalities

A double inequality shows that a value lies between two limits.

 

For example:

$$
2 < x \le 7
$$

This means \( x \) is greater than 2 and less than or equal to 7.

 

Double inequalities can also involve expressions.

 

For example:

$$
1 \le 2x + 3 < 9
$$

The same operation must be applied to all three parts of the inequality.

 

First subtract \( 3 \) from every part:

$$
-2 \le 2x < 6
$$

 

Now divide every part by \( 2 \):

$$
-1 \le x < 3
$$

 

 

Forming and Solving a Double Inequality from Words

For example:

“A number multiplied by 3 is at least 6 but less than 15.”

 

This can be written as:

$$
6 \le 3x < 15
$$

 

Divide all parts by \( 3 \):

$$
2 \le x < 5
$$

 

 

Key Points to Remember

Collect all variable terms on one side before solving.
Reverse the inequality sign when dividing or multiplying by a negative number.
For double inequalities, apply each operation to every part of the inequality.

 

Solving these inequalities correctly allows you to describe ranges of values and constraints clearly and accurately.