inequalities

Ivan and Tony have $x and $(x+50) respectively. It is known that half of amount with tony is at least $10 more than the amount with Ivan, and the sum of the amounts with them is greater than $60.
a)find the rang of the amount with Ivan.
b)find the rang of the amount with Tony.

a)
(x + 50)/2 ≥ x + 10 and x + (x + 50) > 60
x + 50 ≥ 2x + 20 and x + x + 50 > 60
-x ≥ -30 and 2x > 10
x ≤ 30 and x > 5
Hence, 5 < x ≤ 30

Ans: The amount with Ivan is greater than $5 and not greater than $30.

b)
5 < x ≤ 30
(5 + 50) < (x + 50) ≤ (30 + 50)
55 < (x + 50) ≤ 80

Ans: The amount with Ivan is greater than $55 and not greater than $80.


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the isosceles triangleABC, where AB=AC and BC is the longest side.If AC=(x+4)cm and BC=18cm,find the range of values of x.
In ΔABC:
(1) BC is the largest side; and
(2) The longest side must be shorter than the sum of the two shorter side.

18 > x + 4 and 2(x + 4) > 18
14 > x and 2x + 8 > 18
x < 14 and 2x > 10
x < 14 and x > 5

Ans: 5 < x < 14