Absolute Value Inequalities

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Key Questions

  • You have to keep in mind that you can get 2 different outcomes.

    For example, we can take the absolute inequality:
    l4x-12l > 8

    In order to get both possible solutions, you need to write two different equations without the absolute value symbols.
    One would be as written:
    And you can simplify this down:

    For the second outcome, you have to take into account the negative values that could come out of an absolute value equation. Because absolute value is the distance of units away from 0, negative matters.
    In order to do this, we can create another equation where we switch the sign around and make one side negative, again taking away the absolute value symbols.
    4x-12<-8 or -(4x-12)>8
    I would recommend using the first one though because sometimes people forget to distribute the negative in the second equation. Either way, it simplifies down to:

    To make sure this is correct, we can check each equation using a number that complies with the rule with our original equation. For example, if x>5, we should use a number that's greater than 5 and not 5 or else the outcome will equal 8. So in this case, we'll use 6.
    l4(6)-12l > 8
    If we used 4, the answer would turn out to be 4>8 which isn't right.
    Therefore, our answer from before is correct.

    Same goes for the other outcome, we can use a number less than 1, so we'll use 0.
    l4(0)-12l > 8
    l0-12l > 8
    l-12l > 8
    If we used 2, the answer would turn out to be 4>8 which isn't right. Therefore, our answer from before is correct.

  • Example

    The solution set of #|x| > -1# is the set of all real numbers.

    I hope that this was helpful.

  • If the absolute value of the expression is less than 0.

    The absolute value means 'distance from zero' therefore it must either be 0 or a positive number. Distance is never measured as a negative number. Since this is the case, you can never have an absolute value be less than 0.

  • The difference between the two is the symbols used.

    In absolute value equations you use an equal sign to show that the two sides are equal. (Ex. 10x+7 = 37)
    In absolute value inequalities you use a greater than or less than sign to show that one side is greater than or less than the other side. (Ex. 13x+9 > 7x+15)

  • If all the terms of the inequality are absolute values, then Yes, but if the inequality contains a term which is not force to an absolute value then No.

    If all terms of an inequality are absolute values then the only way either side could be negative is if the collection of terms on that side contained a subtraction. For example:
    #|a| -|b| < |c|#
    But such an inequality could always be written without the subtraction by adding an amount equal to that being subtracted to each side
    #(|a| -|b| < |c|) -> (|a| < |c| + |b|)#

    Since this is the case one side of the inequality must have an absolute value that is less than (or less than or equal to) some positive number. Lets call this positive number #K#.

    The minimal side of the inequality must be #< (K)# AND #> (-K)# (a compound relationship).

    Note however, if any of the terms are not absolute values, this does not apply. For example:
    #a < |b|#
    is not a compound relationship; #a# is not restricted except by an upper limit of #|b|#.


  • Double-check the answer
    Justin answered · 1 month ago