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When connecting your battery, connect the positive (red) first and the negative (black) second. When removing the battery, disconnect the negative (black) first and the positive (red) second. Been doing this for nearly 30 years. ALWAYS connect the positive post first.
Larger inverters (500 watts and over) must be hard-wired directly to a battery. The cable size depends on the distance between battery and inverter, and will be specified in the Owner's Manual. When connecting the inverter to the battery use the thickest wire available, in the shortest length practical.