# Linear inequality

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.