Combine double and single quotes.
$myQuery = "SELECT col1 , col2 FROM my_table WHERE col_value LIKE '%$my_string_variable%'";
Although I prefer to protect my code from SQL Injection ..
$myQuery = "SELECT col1 , col2 FROM my_table WHERE col_value LIKE '%" . mysql_real_escape_string($my_string_variable) . "%'";
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