What is the most affordable way to get rid of the gap in my front teeth?

This is an ideal case for bonding and reshaping the front teeth.  In a single visit you can reshape the incisal edges so the teeth look straight and fill the gap to remove black triange.There is minimal amount to do and it will make a tremendous improvement.It is possible to do more but if your only concern is the front two teeth that is an easy quick and inexpensive solution.

Bonding is the process of etching the enamel with a dilute acid, flowing liquid resin on to the treated enamel, then adding composite which is like a putty. The composite is shaped, then cured with a special light, then polished. If done right it is imperceptible from a speaking distance. Unfortunately, it does not last more than a few years before, coffee, food, soft drinks, etc begin to darken it. The composite is not as durable as porcelain. Bonding a porcelain veneer to the tooth will accomplish the same thing but lasts longer-at about 2-3 times the cost.

If you bond or veneer the teeth as they are, then the result will not be aesthetically pleasing. The teeth will appear too wide. For a pleasing result, the width and height of the teeth must be proportional. So instead of ruining the proportions of your teeth with bonding, you should move the teeth with Invisalign or ClearCorrect first. The reason the diastema exists in the first place is that you have a low frenulum (that bump of gum tissue between the teeth that connects to the inside of the upper lip). If you reduce the frenulum (frenectomy), then the teeth will not relapse back into their original position after you move them. Once the teeth have moved into proper position with clear braces and frenectomy, you will not necessarily need veneers; some bonding to refine the shapes of your 6 upper front teeth will have a fantastic result.