My car doesn't want to start?

I drive a 2000 Honda accord! I have a brand new battery, just about a month old now! Sometimes the car will start and itll be fine! The times it doesn't, there isn't a clicking as if there was a bad battery, there's just the ever so slight sound of the fuel pump pumping fuel! I checked my connections and everything seems to be perfect! Could it be a solinoid or the started?

Get in the car and leave the door open. Turn the key to on and look up at your dome light. Turn the key to start. If the dome light doesn't dim or flicker, then the starting circuit is open. The 12 volts isn't going from the battery, through the IGN switch, through the Neutral safety switch, to the starter solenoid. This could be a bad fuse, ignition switch or neutral switch.
If there's a flicker or dimming, the system is sending voltage to the starter solenoid, but it hasn't enough power to engage the solenoid, or you would hear at least a click.

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