You can use +/-5V instead of +/-15V but you need to find an opamp that works well to the positive rail. You could even use a single +5V supply by moving the offset -3V to the non-inverting side and using a positive voltage, but then you need an opamp with good rail to rail performance. A lot of rail to rail opamps tend to tail off at the extremes.
The -3V can be replaced with a could of resistors between the rails. You just need to ensure that the parallel combination of the potential divider is the same as the resistor it would replace. See attached.
I have also attached a single supply version, although it could probably do with a better choice of opamp - I just picked one at random.
The circuit is based on the standard differential amplifier which should be in most books. To select the resistors (or at least the ratios) you first need the gain. In you case you are trying to change +/-3V = 6V into 5V so the ratio is 6/5. So the resistors should be in that ratio. I showed 12k & 10k but you could use other combinations.
The -3V is simply because you want to shift -3V to 3V to start at 0V so your input voltage is relative to -3V.
To use a potential divider instead of a -3V supply, as I have shown, you need to keep the same total resistance while providing -3V. So, you want -3V from -5V so the resistors need to be in the ratio 3/2 (3V across one resistor, 2V across the other). The resistor you are replacing is 12k so the two resistors in parallel need to be 12k. A bit of equation juggling and you find 20k and 30k satisfies both requirements.
If you want to do the shift from a positive voltage so you can use a single supply opamp then you put the resistors on the non-inverting input, as shown. However, now the resistor needs to replace 10k and the shift is 2.5V not 3V because the shift is occurring after the scaling of the input voltage by 5/6. So, 20k and 20k satisfies those requirements.
Hopefully that should help you re-calculate for the -4V/4V requirement.
Keith.