Question

*A triangle has sides 42, 4x, and 2x−6. What is the possible range of x?

Please walk me through the question.

Answer 1

we know that

the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side

so

let
a=42
b=4x
c=2x-6

case 1)
a+b> c
42+4x > 2x-6------> 4x-2x > -6-42-----> 2x > -48------> x > -24

case 2)
a+c > b
42+2x-6 > 4x------> 42-6 > 4x-2x-----> 36 > 2x-----> 18 > x----> x < 18

case 3)
b+c > a
4x+2x-6 > 42-----> 6x > 48------> x > 8

therefore
the answer is
the possible range of x is the interval (8,18)