Question

What is the range of possible sizes for side x? _ < x < _

Answer 1

There are two cases: either x is the largest value or 4.1 is the largest value.

First case: x is largest so that means x < 4.1 + 1.3 which becomes x < 5.8

Second case: 4.1 is largest so that means x + 1.3 > 4.1 which becomes x > 2.8

Answer is 2.8 < x < 5.8.

Answer 2

2.8 < x < 5.8

Explanation:

We must apply the Triangle Inequality Theorem which states that for any triangle with sides a, b, and c:

a + b > c

b + c > a

c + a > b

Here, let's arbitrarily denote a as 4.1, b as 1.3, and c as x. So, let's plug these values into the 3 inequalities listed above:

a + b > c ⇒ 4.1 + 1.3 > x ⇒ 5.8 > x

b + c > a ⇒ 1.3 + x > 4.1 ⇒ x > 2.8

c + a > b ⇒ x + 4.1 > 1.3 ⇒ x > -2.8

Look at the last two: clearly if x is greater than 2.8 (from the second one), then it will definitely be greater than -2.8 (from the third), so we can just disregard the last inequality.

Thus, the range of possible sizes for x are:

2.8 < x < 5.8

~ an aesthetics lover