Start by making the simple syrup (I just turn my faucet on hot for a minute and then fill up a measuring cup with 3/4 cup of water, and then pour in the 1 cup sugar and stir it until the sugar dissolves.) If your faucet water doesn't get very hot, you can just microwave the water for a minute to get it to the desired temperature.
In the bottom of a pitcher add the peach slices and the mint leaves and pour half of the simple syrup in. Use a muddler and muddle the peaches and mint together vigorously for about a minute. Add remaining simple syrup.
Now add the ice into the pitcher, pour in rum (if making a mocktail skip this step). Then pour in Lemon soda water, stir it a few times, and you're done! To serve pour mojitos into glass and spoon in a few peach slices and mint sprigs for garnish, add a straw and enjoy!