solving inequality

Question

|x−3|<3x2−9x+2 solve for x

Answer 1

Absolute value disregards the sign of the expression, so the expression may be either the positive or the negative of the absolute value.  That is:

    if   |x| < y

  then either

      x < y

   or

      -x < y

 

For this problem:

 

|x−3|<3x²−9x+2

 

Either:

   (x-3) < 3x² - 9x + 2

    x < 3x² - 9x + 5         [we may add +3 to both sides and retain sense of inequality]

or

   -(x-3) < 3x² - 9x + 2

    -x + 3 < 3x² - 9x + 2

     -x < (3x² - 9x -1)     [we may subtract 3 to both sides and retain sense of inequality]

     x > -(3x² - 9x - 1)    [but to multiply by (-1) reverses the sense of the inequality]

     x > -3x² + 9x + 1     [distribute the (-1)]

 

Answer:

   Either: 

    x < 3x² - 9x + 5 

   or

    x > -3x² + 9x + 1