how to solve system of inequalities without graph?

First of all, we have to solve the 2 equations:

x+y=4
y=2x-3
ie. x=7/3 y = 5/3

x + y ≥ 4 --- (1)
y ≥ 2x - 3 --- (2)

Therefore, we could divide into 2 ranges:

a) x <= 7/3
(1) y ≥ 4 - x ≥ 4 - 7/3 = 5/3. Since maximum of x is bounded, y ≥ 4 -x is valid in this range of x.
(2) y ≥ 2x - 3, since maximum of x is 7/3, maximum of y is 2(7/3) - 3 = 5/3

Therefore y ≥ 4 - x for all x <= 7/3

b) x > 7/3
(1) x + y ≥ 4
y ≥ 4 - x, since minimum of x is > 7/3, minimum of y ≥ 4 - x > 4 - 7/3 = 5/3
y ≥ 2x - 3 > 2(7/3) - 3 = 5/3. Since minimum of x is bounded, y ≥ 2x-3 is valid in this range of x.

Therefore, x + y ≥ 4 when x <= 7/3
y ≥ 2x - 3 when x ≥ 7/3