Linear inequalities with two variables

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.

  1. We transform inequalities into equality.
  2. We graph the lines separately.
  3. 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:

figure 1

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:

linear inequality example 1

  • 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:

graph3

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:

linear inequality example 2

  • Third inequality: x> - 2

Step 1: we transform it into an equality:

x = - 2

Step 2: we graph the line:

graphic5

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

linear inequality example 3

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:

linear inequality example 4