Determine the numerical value of the current through the 6 ohm resistor by using Thevenin analysis.
First, pull out the element of interest and replace with open terminals.
Redraw the circuit to visualize the voltage differences. The Thevenin voltage will be the open-circuit voltage between the terminals.
The 10 ohm resistor has a 5 V difference across it since it is connected in parallel with a 5 V source. The 3 ohm resistor has a 6 V difference across it since the 2 A from the current source must flow through it (consider KCL at the bottom node). Since the two voltage differences have opposite polarity, the open-circuit voltage is 1 V oriented as shown.
The quickest way to find the Thevenin resistance is to kill the sources. The current source is opened and the voltage source is shorted. The 10 ohm resistor is thereby shorted out and the only resistance seen back through the terminals is the 3 ohm resistance.
The Thevenin generator is shown in the shaded region below. The circuit is now solved by connecting the 6 ohm resistor and finding a current flow of Ix = -0.11 A.