# Solving linear inequalities

**Solving linear inequalities**

Solving linear inequalities are solved similarly to solving equations, however one difference is that both sides can only be multiplied or divided by a **positive** value. For example:

3*x* + 4 ≥ 9 – 2*x* (subtract 4 on both sides)

3*x* ≥ 5 – 2*x *(add 2*x *to both sides)

5*x* ≥ 5 (divided both sides by 5)

*x* ≥ 1

**Multiplying or dividing by negatives**

When multiplying or dividing inequalities by negatives, careful consideration should be taken into account. This is because when an inequality is divided by a negative, the signs are reversed. For example:

–4 < 6 (divide both sides by -1)

4 > -6

**Solving combined linear inequalities**

There are certain circumstances where two inequalities can be written as a single combined inequality. However, two combine there must be some kind of common inequality. For example:

Combine the following:

5*x* + 3 ≥ 5 and 5*x* + 3 < 15

The combined inequality is as follows:

5 ≤ 4*x* + 3 < 15

This is solved using the following steps:

5 ≤ 4*x* + 3 < 15 (subtract 3 from each part)

2 ≤ 4*x* < 12 (divide each part by 4)

0.5 ≤ *x* < 3

**Overlapping solutions**

When solving overlapping inequalities, it should be split into two separate inequalities and simplified. For example:

2*x* – 1 ≤ *x* + 2 < 7 (split the inequalities)

2*x* – 1 ≤ *x* + 2 and *x* + 2 < 7 (simplify the inequalities)

*x* – 1 ≤ 2 and *x* < 5 (simplify further)

*x* ≤ 3 and *x* < 5