In order to graph

h(x) = |2x - 3| - |x + 1|

first consider the graphs of 

f(x) = |2x - 3|, a v-shaped graph with vertex (³/₂,0)

f(x) = -2x + 3 for x ≤ ³/₂ and f(x) = 2x - 3 for x ≥ ³/₂

g(x) = |x + 1|, a v-shaped graph with vertex at (-1,0)

g(x) = -x - 1 for x ≤ -1 and g(x) = x + 1 for x ≥ -1

Then h(x) will behave in the following way on 3 distinct intervals, which are determined by the vertices:

h(x) = (-2x + 3) - (-x - 1) = -x + 4 for x ≤ -1

h(x) = (-2x + 3) - (x + 1) = -3x + 2 for -1 ≤ x ≤ ³/₂

h(x) = (2x - 3) - (x + 1) = x - 4 for x ≥ ³/₂

In the figure below, f(x) is blue, g(x) is red, and h(x) is green.
Be aware that in this figure the grid lines show ½ units, not whole units.

I also recommend that you take a look at Desmos, a free online graphing calculator https://www.desmos.com/calculator/s4ivcdyrgs. Enter and graph a few other absolute value functions on Desmos until you get the hang of it. Good luck!

Thanks!!

You're welcome, Alex! 👍