The inequalities you wrote are mostly correct, but there is a minor error in the last inequality for limited space.
1. The cost inequality is correct: 7x + 5y <= 70 (since Joe wants to spend at most $70)
2. The space inequality is correct: 3x + 15y <= 60 (since Joe has 60 cu inches of space left)
However, the inequality for limited candy purchases (x <= 10) is unnecessary since it is already accounted for in the cost inequality (7x + 5y <= 70).
The last inequality for limited phone card purchases should be y <= 4, as you correctly mentioned (since Joe wants to spend at most $70 and each phone card costs $5).
So the correct system of inequalities to model Joe's possible purchases is:
7x + 5y <= 70 (cost)
3x + 15y <= 60 (space)
y <= 4 (phone cards)