Answer:
1 ) 7 < x < 11
2) 1 < x < 5
3) 1 < x < 11
4) 5 < x < 13
Explanation:
Triangle inequality theorem : the sum of any two sides is always greater than the third side
Let the unknown side be x nad let the known sides by a and b
We will have three inequalities:
(i) a + b > x
(ii) a + x > b
⇒ x > b - a
This will be negative if b < a
(iii) b + x > a
⇒ x > a - b
This will be negative if a < b
From (ii) and (iii)
if a < b, (b-a) will be positive and x > (b - a)
otherwise x > (a - b)
By combining, we can say that x > |a - b|
So our inequalities will be:
(a + b)> x and x > |a - b|
⇒|a - b| < x < (a + b)
This is the required range
1) a = 9, b = 2
|a - b| < x < (a + b)
⇒ |9 - 2| < x < (9 + 2)
⇒ |7| < x < 11
⇒ 7 < x < 11
2) a = 2, b = 3
|a - b| < x < (a + b)
⇒ |2 - 3| < x < (2 + 3)
⇒ |-1| < x < 5
⇒ 1 < x < 5
3) a = 5, b = 6
|a - b| < x < (a + b)
⇒ |5 - 6| < x < (5 + 6)
⇒ |-1| < x < 11
⇒ 1 < x < 11
4) a = 4, b = 9
|a - b| < x < (a + b)
⇒ |4 - 9| < x < (4 + 9)
⇒ |-5| < x < 13
⇒ 5 < x < 13