Question 172028
<pre>
|2x-5|/|2x-1|<1/2 I need you to list solution in interval notation pls. Tnx


We get 0 on the right by subtracting 2 from both sides:

{{{abs(2x-5)/abs(2x-1)<1/2}}}

We first find the endpoints by solving the equation:

{{{abs(2x-5)/abs(2x-1)=1/2}}}

Which involves solving two equations:

{{{(2x-5)/(2x-1)=1/2}}} and {{{(2x-5)/(2x-1)=-1/2}}} 

The first gives solution {{{x=9/2}}} and the second gives
solution 11/6

{{{9/2}}} is about 4.5 and {{11/6}}} is about 1.8333···

Mark those on a number line:

-------------------------o---------------o---------
-2     -1     0     1     2     3     4     5     6 

Test a value to the left of {{{11/6}}}. The easiest
value is 0.  Substitute it into the original
inequality:

{{{abs(2x-5)/abs(2x-1)<1/2}}}
{{{abs(2(0)-5)/abs(2(0)-1)<1/2}}}
{{{abs(-5)/abs(-1)<1/2}}}
{{{5/1<1/2}}}
{{{5<1/2}}}

That is clearly false so do not shade the part to the
left of {{{11/2}}}.  So we still have just:

-------------------------o---------------o---------
-2     -1     0     1     2     3     4     5     6 

Now test a value between {{{11/6}}} and {{{9/2}}}.
The easiest value is 2.  Substitute it into the original
inequality:

{{{abs(2x-5)/abs(2x-1)<1/2}}}
{{{abs(2(2)-5)/abs(2(2)-1)<1/2}}}
{{{abs(4-5)/abs(4-1)<1/2}}}
{{{abs(-1)/abs(3)<1/2}}}
{{{1/3<1/2}}}

That is true so we do shade the part between
{{{11/6}}} and {{{9/2}}}.  Now we have this:

-------------------------o===============o---------
-2     -1     0     1     2     3     4     5     6

Now we test a value to the right of {{{9/2}}}. The easiest
value is 5.  Substitute it into the original
inequality:

{{{abs(2x-5)/abs(2x-1)<1/2}}}
{{{abs(2(5)-5)/abs(2(5)-1)<1/2}}}
{{{abs(10-5)/abs(10-1)<1/2}}}
{{{abs(5)/abs(9)<1/2}}}
{{{5/9<1/2}}}

That is clearly false so do not shade the part to the
rightt of {{{9/2}}}.

So the graph of the solution is

-------------------------o===============o---------
-2     -1     0     1     2     3     4     5     6

and the interval notation is

{{{(matrix(1,3, 11/6,  ",", 9/2))}}}

Edwin</pre>