The economic literature is pretty thoroughly certain - when you raise the price for something, you decrease relative demand for it.
Only in econ 101 classes. Maybe you should take an intermediate level econ class, where you would would learn something beyond that basic rules of thumb.
Seriously, there is no evidence that lower minimum wages, in the real live world, lead to higher unemployment. The theory behind that is that higher wages result in employers consuming less labor, and that might would be true if employers employed scads of people who weren't really neccessary.
In the real world, employers strive to maximize profits, thus they never have "extra" employees, above and beyond what is necessary to maximize profits, thus reducing the number of employees due to higher wages costs would result in lower net profits, as companies would not be able to meet demand, and would litterally be turning away customers who are willing to pay whatever price the company charges for it's goods and services. Thats not a rational behavior to any business owner.
Additionally, the increase in demand created by higher wages would tend to increase company profits, thus offsetting any cost push inflation, and the decrease in per unit operating cost due to economy of scale also servce to reduce cost push inflation.
Maybe you should do some research on the economies of countries like Germany, Australia, and most Scandinavian countries, which all have lower unemployment that we do in the US, similar standards of living to our in the US, lower poverty rates, yet they all have either a government mandated, or socially mandated higher minimum wage. In Australia, it's a government mandated minimum wage, and it's almost double that of the US.
One of my issues with typical conservative economic platforms is that they tend to rely on economic theory, which is cherry picked to match their ideology, instead of real life economic data.