An inequality is an inequality in which there are one or more unknown quantities (unknowns) and what is only verified for certain values of the unknowns. A **linear inequality** (or of **first grade**) only involves adding and subtracting variables to the first power, such as:

ax + by ≥ c

With constant a, b and c and a ≠ 0; b ≠ 0; x and y unknowns. This expression is a **linear inequality with two variables**.

**Solving linear inequalities with two variables**

We will call **linear inequality system**, to the set of values that satisfy (or verify) the inequalities. How do we solve this type of systems? We must **find all pairs of values of x and y for which the inequality holds**. Their solution is one of the half planes that results from representing the resulting equation, which is obtained by transforming the inequality into an equality.

- We transform inequalities into equality.
- We graph the lines separately.
- We take a point at random that satisfies each inequality. For example we take the point (0,0) and we substitute them in them; if this is true, the solution is the half plane where the point is, otherwise the solution will be the other half plane.

**Example: **Solve the following system of inequalities:

4x - 2y ≥ 4

x + y ≤ 1

x> - 2

- First inequality: 4x - 2y ≥ 4

__Step 1__: we transform it into an equality:

4x - 2y = 4

__Step 2__: we graph the line:

4x - 2y = 4 »y = 2 (x - 1).

We give one of the two variables two values, with which we obtain two points:

When x = 0; y = 2 (0 - 1) = - 2

When x = 1; y = 2 (1 - 1) = 0

By representing and joining these two points (0, -2) and (1,0), we obtain a line:

__Step 3__: we take a point at random that satisfies the inequality. For example we take the point (0,0) and substitute them in the inequality; if this is true, the solution is the half plane where the point is, otherwise the solution will be the other half plane:

4x - 2y ≥ 4

4 (0) - 2 (0) ≥ 4

0 ≥ 4 The inequality is not satisfied!

Therefore, the **points of half plane 2 are part of the solution**, that is to say:

- Second inequality: x + y ≤ 1

__Step 1__: we transform it into an equality:

x + y = 1

__Step 2__: we graph the line:

x + y = 1 »y = 1 - x.

We give one of the two variables two values, with which we obtain two points:

When x = 0; y = 1

When x = 1; y = 0

By representing and joining these two points (0, 1) and (1,0), we obtain a line:

__Step 3__: we take a point at random that satisfies the inequality. For example we take the point (0,0):

x + y ≤ 1

0 + 0 ≤ 1

0 ≤ The inequality is not satisfied!

Therefore, the **points of half plane 2 are part of the solution**, that is to say:

- Third inequality: x> - 2

__Step 1__: we transform it into an equality:

x = - 2

__Step 2__: we graph the line:

__Step 3__: As x> - 2, half plane 2 will be part of the solution, that is:

Now, the general solution will be all the points on the plane where the three inequalities are satisfied, that is, the intersection of the three solution planes: