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.

linear inequality exampleIn 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

linear inequality example (8, + ∞)

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

  • 2x - 1 ≤ x + 7

2x - x ≤ 7 + 1

x ≤ 8

linear inequality example  (- ∞, 8]

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