The effect of inflation differentials on the relative prices of goods across countries
During an inflationary period the value of the US Dollar has not increased, but decreased. From within the US, goods are becoming more expensive because each US Dollar is worth less. 

Inflation is believed to be caused by an oversupply of money; if there is an oversupply of US Dollars, it will lead to a decrease in its value. If inflation is increasing at a higher rate in the US than in South Africa, the Dollar will be relatively cheaper, making it more desirable. Thus, an increase in the demand for the US Dollar, and a decrease in its supply.