Question

Verify the triangle inequality in the special cases where (a) a and b have the same sign (b) a ≥ 0, b < 0, and a + b ≥ 0

Answer 1

The Triangle inequality theorem holds true when a and b have the same sign and when a ≥ 0, b < 0, and a + b ≥ 0.

Step-by-step explanation:

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's verify this theorem in the given cases:

When a and b have the same sign:

Assume a and b are positive numbers. So, a > 0 and b > 0.

Then, a + b > a and a + b > b

Therefore, a + b > |a| and a + b > |b|

So, the triangle inequality holds true.

When a ≥ 0, b < 0, and a + b ≥ 0:

Assume a is a positive number and b is a negative number.

We know that |-b| is greater than b

So, a + |-b| > a + b

Therefore, a + |b| > a + b

So, the triangle inequality holds true.

Learn more about Triangle Inequality Theorem