Solving inequalities with modulus

In this publication, we will consider what inequalities with a modulus (one or two) are, and also show how to solve them using practical examples.

Note: what is the modulus of a number, we examined in a separate.

Appearance of inequalities

Inequalities with one modulo look like this:

• |x| > 5

  (modulo x is greater than 5)

• |x – 7| < 4
  (modulo x minus 7 is less than 4)
• |x + 2| ≥ 11

  (modulo x plus 2 is greater than or equal to 11)

An unknown variable is specified in the module x or an expression with it.

Solving inequalities

Let’s take a look at the solution of the above examples.

|x| > 5

This should be understood in this way: there are points on the number axis, the distance from which to zero is greater than five. Those. these are points greater than 5 or less than -5, therefore, this inequality has two solutions: x₁ ∈ (-∞; -5) и x₂ ∈ (5; ∞).

|x – 7| < 4

Those. there are so many points on the number line x, the distance from which to point 7 is less than 4. Therefore, the inequality has the following solution:

7 – 4 < x < 7 + 4 or x ∈ (3; 11).

|x + 2| ≥ 11

This inequality can be represented as follows: |x – (-2)| ≥ 11.

Thus, on the numerical axis of the point x are at least 11 away from point -2. Means:

• x₁  ≥ 9 (-2 + 11) or x₁ ∈ [9; ∞)
• x₂ ≤ -13 (-2 – 11) or x₂ ∈ (-∞; -13]

Note:

Some inequalities may contain two modules: for example: |x| > |and|.

It also has two solutions: x₁ ∈ (-∞; -y) и x₂ ∈ (and; ∞).