The question states the following:

Suppose in Figure P1-22 a ground is connected to the minus side of element 6 and another to the junction of elements 2, 3 and 4. Further, assume that the voltage v4 is 5V and v1 is 10V. What are v2, v3, v5 & v6?

I know that voltage is a "through" variable measured in reference to ground (which is 0 V). Does this mean that since v4 is 5V and current is moving towards ground, v6 has to be -5V?

 

I believe you're right, but the polarity signs on the photo seem to be wrong for this set of conditions.

 

Polarity signs are arbitrary. Notice that there is no distinction between loads and sources in the circuit as given, so all of the voltages and currents were assigned on a per element basis using the passive sign convention.

Voltage is not a "through" variable. Voltages appear ACROSS elements and currents pass THROUGH elements.

You don't need ANY of the current indicators to solve this problem.

You are given that the voltage at the junction of the 2-3-4 element set is 0V.

You are given that the voltage across element 4 is 5V. Given the indicated polarity of V4, this means that the node at the junction of the 4-5-6 element set is -5V.

V6 is defined as the voltage at this latter junction minus the voltage at the bottom node, which is also given as 0V. Hence V6 is -5V.

Notice that, with the information given, there is no way to determine if Element 1 is a load and Element 2 is a source or the other way around because there is no way to determine the polarity of the currents through them.