Question 261961
<font face="Garamond" size="+2">


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 1\ >\ 3x]


Is a rather handsome little single variable inequality.  


"How you do it" is really a function of what you want to do with it.  I don't know what the "awnser" is.  I don't know what "awnser" means.  If you meant "answer" then your inequality doesn't have one, per se.  What it has is a solution set that consists of a set of infinitely many real number values which, when substituted for the variable in the inequality, make the inequality a true statement.


You can simplify the inequality and thereby derive a description of the values that are part of the solution set.  If that is what you meant by "do it" in the sense of "How do you do it", then I may be able to help you:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ 1\ >\ 3x]


Add *[tex \Large -3x] to both sides of the inequality:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -x\ +\ 1\ >\ 0]


Add -1 to both sides of the inequality:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -x\ >\ -1]


Multiply by -1.  Remember that when you multiply an inequality by a negative number, you must reverse the sense of the inequality -- > becomes < and < becomes >.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ <\ 1]


So our solution set is the set of all real numbers that are less than 1.


Check the answer.  Pick a number less than 1, let's choose 0.  Substitute 0 into the original inequality:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2(0)\ +\ 1\ >\ 3(0)]


which is to say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 1\ >\ 0]


which is a true statement, and gives us a clue that we are on the right track.


Now pick a number greater than 1.  Let's try 2.  Again, substitute:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2(2)\ +\ 1\ >\ 3(2)]


which is to say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 5\ >\ 6]


which is a false statement as it should be since we selected a value that is not in the solution set.


And if you really want to make sure, pick 1 and substitute:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2(1)\ +\ 1\ >\ 3(1)]


which is to say:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3\ >\ 3]


which is also a false statement.  3 is equal to 3, not greater than 3.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
</font>