I had a link with text and an image. I don’t want the image to have an underline under it but I do want the text to have an underline. Is there anyway to support this without breaking these down into two seperate links? Here is an example picture:

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No, there is no way to achieve that without splitting the link into two parts.
The alternative is to put
border-bottom: 1px solid #000on to the element containing the text, however this will not behave in the same was as a native A element, with hover, active and visited states – at least not without the intervention of javascript.