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Question

Absolute values inequalities are like compound inequalties. A compound inequaltity cosists of 2 inequalities possibly separated by a word. A conjunction compound inequaltiy uses the word AND while a disjunction compound inequality uses OR. Which 2 absolute value inequalities are similar to conjunction (AND) compound inequalities and which 2 absolute value in equalities are similar to disjunction (OR) compound inequalities?

Answer 1

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Answer 2

Let's work through the problem in reverse order:

 

Compound inequality examples:

     Conjunction (AND):     x > -5 AND x < 5     means  -5 < x < 5

     Disjunction (OR):        x < -5  OR  x > 5    means  x < -5 OR x > 5

 

Now, to use absolute value inequalities:

     |x| < y   means    -y < x < y

               which is  x> -y  AND  x < y

               [notice that y is an upper limit to the magnitude (absolute value)]

 

     |x| > y   means    -x > y  OR   +x > y

               [notice that y is a lower limit to the magnitude (absolute value)]

  

The other two, "≤" and "≥", are similar.

 

So, matching the answers to the problem:

     |x|<y and |x|≤y  are similar to conjunctive (AND) compound inequalities

   and

     |x|>y and |x|≥y  are similar to disjunctive (OR) compound inequalities

 

 

Application:

When solving absolute value inequalities, you may have two parts to the answer.  For example:

         |x-3| > 10

    means

       either   -(x-3) > 10

                    -x + 3 > 10    [distribute -1]

                      -x > 7          [subtract +3 from both sides retains sense of inequality]

                       x < -7         [multiply both sides by (-1) reverses sense of inequality]

          OR    +(x-3) > 10

                    x - 3 > 10      [distribute +1]

                    x > 13           [add +3 to both sides retains sense of inequality]

Answer 3

Conjunction:

 

2 < x < 5 same as 2 < x and x < 5

 

Disjunction:

 

x < 2 or x > 5