A **linear inequality** (or of **first grade**) of a variable can be written in the following ways:

- ax <b
- ax> b
- ax ≤ b
- ax ≥ b

With a and b constant with a ≠ 0; b ≠ 0; x unknown.

In the case of an inequality of one or more variables, it can be written as:

- ax + by <c
- ax + by> c
- ax + by ≤ c
- ax + by ≥ c

With constant a, b and c and with a ≠ 0; b ≠ 0; x and y unknowns.

**Properties**

**Equivalence criteria**

- If the same number is added or subtracted to the two members of an inequality, the resulting inequality is equivalent to the one given.
- If the two members of an inequality are multiplied or divided by the same positive number, the resulting inequality is equivalent to the one given.
- If the two members of an inequality are multiplied or divided by the same negative number, the resulting inequality changes direction and is equivalent to the one given.

**Resolution of linear inequalities**

The solution of an inequality is the set of values of the variable that verifies the inequality. We can express the solution of the inequality by:

- A graphic representation.
- An interval.

**Examples: **Solve the following linear inequalities with an unknown

- 2x - 1> x + 7

Passing x to the first member

2x - 1 - x> 7

Now passing 1 to the second member

2x - x> 7 + 1

Reducing, we have

x> 8

(8, + ∞)

Inequality is only **check** for the values of **x greater than 8**.

- 2x - 1 ≤ x + 7

2x - x ≤ 7 + 1

x ≤ 8

(- ∞, 8]

Inequality is only **check** for the values of **x less than or equal to 8**.