<PRE><B>Question 1116383</B>
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In the first inequality, there are two absolute value expressions, so we have to consider different cases when x is positive or negative and when x-2 is positive or negative.  So we have three intervals on the number line to consider: from negative infinity to 0; from 0 to 2, and from 2 to infinity.&lt;br&gt;
The right side of the inequality makes the first interval easy.  For negative values of x, {{{x+abs(x)=0}}}.  The inequality then says that some absolute value is greater than 0, which is always true.&lt;br&gt;
So all negative numbers are part of the solution set of the inequality.&lt;br&gt;
Between 0 and 2, {{{abs(x-2)=2-x}}} so the inequality says
{{{2-x &gt; x+x}}}
{{{2-x &gt; 2x}}}
{{{2 &gt; 3x}}}
{{{x &lt; 2/3}}}&lt;br&gt;
So, of the x values between 0 and 2, the ones that satisfy the inequality are those less than 2/3.&lt;br&gt;
For the interval from 2 to infinity, {{{abs(x-2) = x-2}}} and the inequality says
{{{x-2 &gt; x+x}}}
{{{x-2 &gt; 2x}}}
{{{x &lt; -2}}}&lt;br&gt;
But this apparent solution is not in the interval we are working in.  So no values of x in the interval from 2 to infinity satisfy the inequality.&lt;br&gt;
So our final solution set is from negative infinity to 2/3, not including 2/3.&lt;br&gt;
The second inequality is a bit easier, because there is only one absolute value expression.  {{{abs(4x-1)}}} is equal to 4x-1 for x values greater than or equal to 1/4, or equal to 1-4x for x values less than 1/4.  So&lt;br&gt;
(a) for x &lt; 1/4,
{{{1-4x &gt; x-2}}}
{{{3 &gt; 5x}}}
{{{x &lt; 3/5}}}&lt;br&gt;
ALL values of x that are less than 1/4 are less than 3/5; so all values of x less than 1/4 are solutions to the inequality.&lt;br&gt;
(b) for x &gt; 1/4,
{{{4x-1 &gt; x-2}}}
{{{3x &gt; -1}}}
{{{x &gt; -1/3}}}&lt;br&gt;
Again, all values of x that are greater than 1/4 are greater than -1/3.&lt;br&gt;
So every value of x is a solution to the second inequality.&lt;br&gt;
This is easily verified with even a rough sketch of the graphs of the two expressions:&lt;br&gt;
{{{graph(400,400,-3,5,-3,5,abs(4x-1),x-2)}}}&lt;br&gt;
The value of the absolute value expression is always greater than the value of the linear expression.</PRE>