document.write( "Question 1116383: Find the set of values of x which satisfies each of the following inequalities.
\n" ); document.write( "a) |x - 2| > x + |x|,
\n" ); document.write( "b) |4x - 1| > x - 2 \n" ); document.write( "

Algebra.Com's Answer #731278 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "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.

\n" ); document.write( "The right side of the inequality makes the first interval easy. For negative values of x, $\"x%2Babs%28x%29=0\"$ . The inequality then says that some absolute value is greater than 0, which is always true.

\n" ); document.write( "So all negative numbers are part of the solution set of the inequality.

\n" ); document.write( "Between 0 and 2, $\"abs%28x-2%29=2-x\"$ so the inequality says
\n" ); document.write( " $\"2-x+%3E+x%2Bx\"$
\n" ); document.write( " $\"2-x+%3E+2x\"$
\n" ); document.write( " $\"2+%3E+3x\"$
\n" ); document.write( " $\"x+%3C+2%2F3\"$

\n" ); document.write( "So, of the x values between 0 and 2, the ones that satisfy the inequality are those less than 2/3.

\n" ); document.write( "For the interval from 2 to infinity, $\"abs%28x-2%29+=+x-2\"$ and the inequality says
\n" ); document.write( " $\"x-2+%3E+x%2Bx\"$
\n" ); document.write( " $\"x-2+%3E+2x\"$
\n" ); document.write( " $\"x+%3C+-2\"$

\n" ); document.write( "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.

\n" ); document.write( "So our final solution set is from negative infinity to 2/3, not including 2/3.

\n" ); document.write( "The second inequality is a bit easier, because there is only one absolute value expression. $\"abs%284x-1%29\"$ 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

\n" ); document.write( "(a) for x < 1/4,
\n" ); document.write( " $\"1-4x+%3E+x-2\"$
\n" ); document.write( " $\"3+%3E+5x\"$
\n" ); document.write( " $\"x+%3C+3%2F5\"$

\n" ); document.write( "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.

\n" ); document.write( "(b) for x > 1/4,
\n" ); document.write( " $\"4x-1+%3E+x-2\"$
\n" ); document.write( " $\"3x+%3E+-1\"$
\n" ); document.write( " $\"x+%3E+-1%2F3\"$

\n" ); document.write( "Again, all values of x that are greater than 1/4 are greater than -1/3.

\n" ); document.write( "So every value of x is a solution to the second inequality.

\n" ); document.write( "This is easily verified with even a rough sketch of the graphs of the two expressions:

\n" ); document.write( " $\"graph%28400%2C400%2C-3%2C5%2C-3%2C5%2Cabs%284x-1%29%2Cx-2%29\"$

\n" ); document.write( "The value of the absolute value expression is always greater than the value of the linear expression. \n" ); document.write( "

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