document.write( "Question 1108359: Solve :
\n" ); document.write( "1/|2x-1| greater than or equal to 1 \n" ); document.write( "
Algebra.Com's Answer #723369 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "Solve: \"1%2Fabs%282x-1%29%3C=1\"

\n" ); document.write( "I'll show a couple of different ways you could solve this. You should know and understand both methods; in any given problem one or the other might be easier to use.

\n" ); document.write( "And there are undoubtedly other methods; perhaps you will get answers from other tutors that show you method(s) that are different than these two.

\n" ); document.write( "(1) One thing you can do is determine the values of x for which the expression is undefined and the values for which equality holds. That will divide the number line into segments; you can then check in which of those segments of the number line the inequality is satisfied.

\n" ); document.write( "The left side of the inequality is undefined when the denominator is 0; that is at x = 1/2.

\n" ); document.write( "The equation \"1%2Fabs%282x-1%29=1\" is satisfied when \"abs%282x-1%29=1\", which is when x = 0 and when x = 1.

\n" ); document.write( "So the number line is divided into the segments
\n" ); document.write( "(-infinity, 0], [0,1/2), (1/2,1], and [1, infinity).

\n" ); document.write( "Picking values in each of these segments shows the inequality is satisfied on [0,1/2) and (1/2,1].


\n" ); document.write( "(2) A more traditional algebraic approach is to separate the solution into two cases, depending on whether 2x-1 is positive or negative. (We already know we don't need to check the case where 2x-1 is 0, because that makes the inequality invalid.)

\n" ); document.write( "(a) If \"x+%3E+1%2F2\" then \"2x-1%3E0\"; then \"1%2Fabs%282x-1%29+=+1%2F%282x-1%29\" and the inequality is
\n" ); document.write( "\"1%2F%282x-1%29+%3E=+1\"
\n" ); document.write( "\"1+%3E=+2x-1\"
\n" ); document.write( "\"2+%3E=+2x\"
\n" ); document.write( "\"x+%3C=+1\"

\n" ); document.write( "So in this case, where we are only considering values of x greater than 1/2, the solution set is all numbers less than or equal to 1; that gives us the (1/2,1] part of the solution.

\n" ); document.write( "(b) If \"x+%3C+1%2F2\" then \"2x-1%3C0\"; then \"1%2Fabs%282x-1%29=+1%2F%281-2x%29\" and the inequality is
\n" ); document.write( "\"1%2F%281-2x%29+%3E=+1\"
\n" ); document.write( "\"1+%3E=+1-2x\"
\n" ); document.write( "\"2x+%3E=+0\"
\n" ); document.write( "\"x+%3E=+0\"

\n" ); document.write( "So in this case, where we are only considering values of x less than 1/2, the solution set is all numbers greater than or equal to 0; that gives us the [0,1/2) part of the solution.


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\n" ); document.write( "I don't know why tutor teachmath bothered to give you an incorrect answer without showing any work.... That's a strange way of interpreting \"teach math\".

\n" ); document.write( "The answer that tutor shows is [0,.5)U[.5,1]. There are two things wrong with that answer:
\n" ); document.write( "(1) x=.5 is not included in the solution set, as shown in the second interval of the answer.
\n" ); document.write( "(2) the answer is equivalent to [0,1] \n" ); document.write( "
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