problems on linear inequalities

Question

find the smallest three consecutive  integers in which the difference between one fifth of the largest and one third of the smallest is at least 3

Answer 1

Let the integers be x-1, x and x+1.

(x+1)/5-(x-1)/3≥3.

Multiply through by 15:

3x+3-5x+5≥45.

-2x+8≥45,

8-45≥2x,

-37≥2x,

-18.5≥x, x≤-18.5. There are many solutions for x so a minimum cannot be defined.

However, if (x-1)/3-(x+1)/5≥3,

5x-5-3x-3≥45

2x-8≥45,

x-4≥22.5

x≥26.5. Minimum value is x=27, the integers are 26, 27, 28.

CHECK

28/5=5.6, 26/3=8.67. Difference is 3.07 approx.