TRIANGLE INEQUALITY: PROOF, EXAMPLES, SOLVED EXERCISES - MATHS - 2025

It is called unequal triangle property that meet two real numbers consisting of the absolute value of their sum is always less than or equal to the sum of their absolute values. This property is also known as Minkowski's inequality or triangular inequality.

This property of numbers is called triangular inequality because in triangles it happens that the length of one side is always less than or equal to the sum of the other two, even though this inequality does not always apply in the area of ​​triangles.

Figure 1. The absolute value of the sum of two numbers is always less than or equal to the sum of their absolute values. (Prepared by R. Pérez)

There are several proofs of the triangular inequality in real numbers, but in this case we will choose one based on the properties of the absolute value and the binomial squared.

Theorem: For every pair of numbers a and b belonging to the real numbers we have:

- a + b - ≤ - a - + - b -

Demonstration

We begin by considering the first member of the inequality, which will be squared:

- a + b - ^ 2 = (a + b) ^ 2 = a ^ 2 + 2 ab + b ^ 2 (Eq. 1)

In the previous step we used the property that any number squared is equal to the absolute value of said squared number, that is: -x- ^ 2 = x ^ 2. The square binomial expansion has also been used.

Every number x is less than or equal to its absolute value. If the number is positive it is equal, but if the number is negative it will always be less than a positive number. In this case its own absolute value, that is, it can be stated that x ≤ - x -.

The product (ab) is a number, therefore it applies that (ab) ≤ - ab -. When this property is applied to (Eq. 1) we have:

- a + b - ^ 2 = a ^ 2 + 2 (ab) + b ^ 2 ≤ a ^ 2 + 2 - ab - + b ^ 2 (Eq. 2)

Taking into account that - ab - = - a - b - la (Eq. 2) can be written as follows:

- a + b - ^ 2 ≤ a ^ 2 + 2 - a - b - + b ^ 2 (Eq. 3)

But since we said before that the square of a number is equal to the absolute value of the number squared, then equation 3 can be rewritten as follows:

- a + b - ^ 2 ≤ -a- ^ 2 + 2 -a- -b- + -b- ^ 2 (Eq. 4)

In the second member of the inequality, a remarkable product is recognized, which when applied leads to:

- a + b - ^ 2 ≤ (-a- + -b -) ^ 2 (Eq. 5)

In the previous expression it should be noted that the values ​​to be squared in both members of the inequality are positive, therefore it must also be satisfied that:

- a + b - ≤ (-a- + -b-) (Eq. 6)

The previous expression is exactly what you wanted to demonstrate.

Examples

Next we will check the triangular inequality with several examples.

Example 1

We take the value a = 2 and the value b = 5, that is, both positive numbers and we check whether the inequality is satisfied or not.

- 2 + 5 - ≤ -2- + -5-

- 7 - ≤ -2- + -5-

7 ≤ 2+ 5

Equality is verified, therefore the triangle inequality theorem has been fulfilled.

Example 2

The following values ​​a = 2 and b = -5 are chosen, that is, a positive number and the other negative, we check whether or not the inequality is satisfied.

- 2 - 5 - ≤ -2- + --5-

- -3 - ≤ -2- + --5-

3 ≤ 2 + 5

The inequality is satisfied, therefore the triangular inequality theorem has been verified.

Example 3

We take the value a = -2 and the value b = 5, that is, a negative number and the other positive, we check whether or not the inequality is satisfied.

- -2 + 5 - ≤ --2- + -5-

- 3 - ≤ --2- + -5-

3 ≤ 2 + 5

The inequality is verified, therefore the theorem has been fulfilled.

Example 4

The following values ​​a = -2 and b = -5 are chosen, that is, both negative numbers and we check whether or not the inequality is satisfied.

- -2 - 5 - ≤ --2- + --5-

- -7 - ≤ --2- + --5-

7 ≤ 2+ 5

Equality is verified, therefore Minkowski's inequality theorem has been fulfilled.

Example 5

We take the value a = 0 and the value b = 5, that is, a number zero and the other positive, then we check whether the inequality is satisfied or not.

- 0 + 5 - ≤ -0- + -5-

- 5 - ≤ -0- + -5-

5 ≤ 0+ 5

Equality is fulfilled, therefore the triangle inequality theorem has been verified.

Example 6

We take the value a = 0 and the value b = -7, that is to say a number zero and the other positive, then we check whether the inequality is satisfied or not.

- 0 - 7 - ≤ -0- + --7-

- -7 - ≤ -0- + --7-

7 ≤ 0+ 7

Equality is verified, therefore the triangular inequality theorem has been fulfilled.

Solved exercises

In the following exercises, represent geometrically the triangle inequality or Minkowski inequality for the numbers a and b.

The number a will be represented as a segment on the X axis, its origin O coincides with the zero of the X axis and the other end of the segment (at point P) will be in the positive direction (to the right) of the X axis if a > 0, but if a <0 it will be towards the negative direction of the X axis, as many units as its absolute value indicates.

Similarly, the number b will be represented as a segment whose origin is on point P. The other extreme, that is, point Q will be to the right of P if b is positive (b> 0) and point Q will be -b - units to the left of P if b <0.

Exercise 1

Graph the inequality of the triangle for a = 5 and b = 3 - a + b - ≤ - a - + - b -, where c = a + b.

Exercise 2

Graph the triangular inequality for a = 5 and b = -3.

- a + b - ≤ - a - + - b -, where c = a + b.

Exercise 3

Graphically show the inequality of the triangle for a = -5 and b = 3.

- a + b - ≤ - a - + - b -, where c = a + b.

Exercise 4

Graphically construct the triangular inequality for a = -5 and b = -3.

- a + b - ≤ - a - + - b -, where c = a + b.

References

1. E. Whitesitt. (1980). Boolean Algebra and its Applications. Editorial Company Continental CA
2. Mícheál O 'Searcoid. (2003) Elements of Abstract Analysis.. Department of mathematics. University college Dublin, Beldfield, Dublind.
3. J. Van Wyk. (2006) Mathematics and Engineering in Computer Science. Institute for Computer Sciences and Technology. National Bureau of Standards. Washington, DC 20234
4. Eric Lehman. Mathematics for Computer Science. Google Inc.
5. F Thomson Leighton (1980). Calculus. Department of Mathematics and the Computer Science and AI Laboratory, Massachussetts Institute of Technology.
6. Khan Academy. Triangle Inequality Theorem. Recovered from: khanacademy.org
7. Wikipedia. Triangular inequality. Recovered from: es. wikipedia.com